Properties

Label 4-165e2-1.1-c3e2-0-2
Degree $4$
Conductor $27225$
Sign $1$
Analytic cond. $94.7763$
Root an. cond. $3.12014$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 6·3-s − 11·4-s − 10·5-s − 6·6-s − 4·7-s + 15·8-s + 27·9-s + 10·10-s − 22·11-s − 66·12-s − 90·13-s + 4·14-s − 60·15-s + 61·16-s − 16·17-s − 27·18-s − 170·19-s + 110·20-s − 24·21-s + 22·22-s − 124·23-s + 90·24-s + 75·25-s + 90·26-s + 108·27-s + 44·28-s + ⋯
L(s)  = 1  − 0.353·2-s + 1.15·3-s − 1.37·4-s − 0.894·5-s − 0.408·6-s − 0.215·7-s + 0.662·8-s + 9-s + 0.316·10-s − 0.603·11-s − 1.58·12-s − 1.92·13-s + 0.0763·14-s − 1.03·15-s + 0.953·16-s − 0.228·17-s − 0.353·18-s − 2.05·19-s + 1.22·20-s − 0.249·21-s + 0.213·22-s − 1.12·23-s + 0.765·24-s + 3/5·25-s + 0.678·26-s + 0.769·27-s + 0.296·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(94.7763\)
Root analytic conductor: \(3.12014\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{165} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 27225,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 - p T )^{2} \)
5$C_1$ \( ( 1 + p T )^{2} \)
11$C_1$ \( ( 1 + p T )^{2} \)
good2$D_{4}$ \( 1 + T + 3 p^{2} T^{2} + p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 4 T + 622 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 90 T + 6402 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 16 T + 1662 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 170 T + 994 p T^{2} + 170 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 124 T + 12878 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 158 T + 50106 T^{2} + 158 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 60 T + 59870 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 372 T + 82590 T^{2} + 372 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 38 T + 37410 T^{2} - 38 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 12 p T + 160230 T^{2} + 12 p^{4} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 224 T + 466 p T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 472 T + 190182 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 248 T + 181334 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 72 T - 107850 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 744 T + 738822 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 2060 T + 1768494 T^{2} - 2060 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 486 T + 822786 T^{2} + 486 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 642 T + 691166 T^{2} - 642 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 286 T + 392750 T^{2} + 286 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 244 T + 1355190 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 168 T + 1053870 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.23792679064640545073851997807, −11.85094862469132100314779326842, −10.90404551148103832380410403804, −10.37622224819429722673356132702, −9.908130172095537346889267006193, −9.597167129720741986027704417040, −8.894333398709967078008516095039, −8.541040759243852194573333948699, −8.072731700173424179396919041230, −7.73987841290595667377532682041, −7.03109962557393122411992906109, −6.46857242033609109549585217379, −5.09468681984002093006600297473, −4.94825489858654283715416737940, −3.96145918542717960996069715914, −3.84880495187668241124743405475, −2.67272850484331780433849265635, −1.95852522855644175105115195687, 0, 0, 1.95852522855644175105115195687, 2.67272850484331780433849265635, 3.84880495187668241124743405475, 3.96145918542717960996069715914, 4.94825489858654283715416737940, 5.09468681984002093006600297473, 6.46857242033609109549585217379, 7.03109962557393122411992906109, 7.73987841290595667377532682041, 8.072731700173424179396919041230, 8.541040759243852194573333948699, 8.894333398709967078008516095039, 9.597167129720741986027704417040, 9.908130172095537346889267006193, 10.37622224819429722673356132702, 10.90404551148103832380410403804, 11.85094862469132100314779326842, 12.23792679064640545073851997807

Graph of the $Z$-function along the critical line