Properties

Label 2-15e2-1.1-c7-0-22
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $70.2866$
Root an. cond. $8.38371$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·2-s − 28·4-s + 1.17e3·7-s + 1.56e3·8-s + 2.65e3·11-s + 1.11e4·13-s − 1.17e4·14-s − 1.20e4·16-s + 3.10e4·17-s + 3.03e4·19-s − 2.65e4·22-s + 3.27e4·23-s − 1.11e5·26-s − 3.27e4·28-s − 1.63e5·29-s + 1.36e5·31-s − 7.95e4·32-s − 3.10e5·34-s − 1.66e4·37-s − 3.03e5·38-s + 4.83e5·41-s + 1.41e5·43-s − 7.42e4·44-s − 3.27e5·46-s − 1.03e5·47-s + 5.45e5·49-s − 3.13e5·52-s + ⋯
L(s)  = 1  − 0.883·2-s − 0.218·4-s + 1.28·7-s + 1.07·8-s + 0.600·11-s + 1.41·13-s − 1.13·14-s − 0.733·16-s + 1.53·17-s + 1.01·19-s − 0.530·22-s + 0.561·23-s − 1.24·26-s − 0.282·28-s − 1.24·29-s + 0.821·31-s − 0.428·32-s − 1.35·34-s − 0.0540·37-s − 0.896·38-s + 1.09·41-s + 0.270·43-s − 0.131·44-s − 0.496·46-s − 0.145·47-s + 0.662·49-s − 0.308·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(70.2866\)
Root analytic conductor: \(8.38371\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.857442140\)
\(L(\frac12)\) \(\approx\) \(1.857442140\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 5 p T + p^{7} T^{2} \)
7 \( 1 - 1170 T + p^{7} T^{2} \)
11 \( 1 - 2650 T + p^{7} T^{2} \)
13 \( 1 - 860 p T + p^{7} T^{2} \)
17 \( 1 - 31070 T + p^{7} T^{2} \)
19 \( 1 - 30316 T + p^{7} T^{2} \)
23 \( 1 - 32760 T + p^{7} T^{2} \)
29 \( 1 + 163150 T + p^{7} T^{2} \)
31 \( 1 - 136188 T + p^{7} T^{2} \)
37 \( 1 + 16640 T + p^{7} T^{2} \)
41 \( 1 - 483200 T + p^{7} T^{2} \)
43 \( 1 - 141080 T + p^{7} T^{2} \)
47 \( 1 + 103240 T + p^{7} T^{2} \)
53 \( 1 + 1950130 T + p^{7} T^{2} \)
59 \( 1 + 2643350 T + p^{7} T^{2} \)
61 \( 1 - 2820922 T + p^{7} T^{2} \)
67 \( 1 - 506220 T + p^{7} T^{2} \)
71 \( 1 + 2890900 T + p^{7} T^{2} \)
73 \( 1 - 2877290 T + p^{7} T^{2} \)
79 \( 1 + 5717556 T + p^{7} T^{2} \)
83 \( 1 - 3790380 T + p^{7} T^{2} \)
89 \( 1 + 10564500 T + p^{7} T^{2} \)
97 \( 1 + 2158130 T + p^{7} T^{2} \)
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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

−10.97861556797155465612588434716, −9.846458301241903245311351265600, −8.980507450008903136312069530719, −8.096725913359239005370087394501, −7.44388234568116609440877897938, −5.82732131739986591307675281551, −4.72732164558773949335512602696, −3.52678647634174601862493783525, −1.50827814221873029889901605308, −0.968149300687189402217265168089, 0.968149300687189402217265168089, 1.50827814221873029889901605308, 3.52678647634174601862493783525, 4.72732164558773949335512602696, 5.82732131739986591307675281551, 7.44388234568116609440877897938, 8.096725913359239005370087394501, 8.980507450008903136312069530719, 9.846458301241903245311351265600, 10.97861556797155465612588434716

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