Properties

Label 2-7350-1.1-c1-0-42
Degree $2$
Conductor $7350$
Sign $1$
Analytic cond. $58.6900$
Root an. cond. $7.66094$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 3·13-s + 16-s + 3·17-s − 18-s + 2·19-s + 3·23-s − 24-s − 3·26-s + 27-s + 7·29-s − 31-s − 32-s − 3·34-s + 36-s + 4·37-s − 2·38-s + 3·39-s + 5·41-s − 3·43-s − 3·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.832·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.458·19-s + 0.625·23-s − 0.204·24-s − 0.588·26-s + 0.192·27-s + 1.29·29-s − 0.179·31-s − 0.176·32-s − 0.514·34-s + 1/6·36-s + 0.657·37-s − 0.324·38-s + 0.480·39-s + 0.780·41-s − 0.457·43-s − 0.442·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(58.6900\)
Root analytic conductor: \(7.66094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.136558085\)
\(L(\frac12)\) \(\approx\) \(2.136558085\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.77909327836352500299774911308, −7.59872898924479954069115411356, −6.55968805080063558805225799482, −6.05257649328496999954777952055, −5.10202905571881481355397918084, −4.22824987861372747254183750412, −3.27862185521101989220592144376, −2.76469986240008805227884851071, −1.61720231748495350569304843286, −0.853104750646525128048413960087, 0.853104750646525128048413960087, 1.61720231748495350569304843286, 2.76469986240008805227884851071, 3.27862185521101989220592144376, 4.22824987861372747254183750412, 5.10202905571881481355397918084, 6.05257649328496999954777952055, 6.55968805080063558805225799482, 7.59872898924479954069115411356, 7.77909327836352500299774911308

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