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Properties

Label 2-325-13.3-c1-0-14
Degree $2$
Conductor $325$
Sign $0.548 + 0.835i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.134 − 0.232i)2-s + (0.301 + 0.522i)3-s + (0.963 − 1.66i)4-s + (0.0810 − 0.140i)6-s + (0.715 − 1.23i)7-s − 1.05·8-s + (1.31 − 2.28i)9-s + (0.0810 + 0.140i)11-s + 1.16·12-s + (−2.41 + 2.67i)13-s − 0.384·14-s + (−1.78 − 3.09i)16-s + (1.41 − 2.44i)17-s − 0.708·18-s + (1.96 − 3.40i)19-s + ⋯
L(s)  = 1  + (−0.0950 − 0.164i)2-s + (0.174 + 0.301i)3-s + (0.481 − 0.834i)4-s + (0.0330 − 0.0573i)6-s + (0.270 − 0.468i)7-s − 0.373·8-s + (0.439 − 0.761i)9-s + (0.0244 + 0.0423i)11-s + 0.335·12-s + (−0.670 + 0.742i)13-s − 0.102·14-s + (−0.446 − 0.773i)16-s + (0.342 − 0.592i)17-s − 0.167·18-s + (0.450 − 0.780i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.548 + 0.835i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (276, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.548 + 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30850 - 0.706297i\)
\(L(\frac12)\) \(\approx\) \(1.30850 - 0.706297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (2.41 - 2.67i)T \)
good2 \( 1 + (0.134 + 0.232i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.301 - 0.522i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.715 + 1.23i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0810 - 0.140i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.41 + 2.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.96 + 3.40i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.36 - 4.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.99 - 3.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.453T + 31T^{2} \)
37 \( 1 + (2.52 + 4.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.29 - 7.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.33 + 4.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 - 7.30T + 53T^{2} \)
59 \( 1 + (4.98 - 8.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.726 - 1.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.17 - 5.50i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.02 - 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.75T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 - 9.45T + 83T^{2} \)
89 \( 1 + (-4.33 - 7.50i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.40 - 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.43021679818646908695468425357, −10.46607022218642748003500768551, −9.605578193179933896889976380921, −9.066455423612529080995609951809, −7.35418158910286248817041685081, −6.79219310570615819398789212897, −5.43033595061470435161501801932, −4.38253418394432217872515640317, −2.88896443726096609350629779134, −1.22249992441489060596500792987, 2.04691440609770714909605607105, 3.21827616564514736408726341938, 4.72953985306795719842202753117, 6.00036447364893754614782741171, 7.19787450661979239667536816647, 7.947163007507300041065290417534, 8.570257900094381620791207790329, 9.970862927052501390850645376729, 10.85277327539404454988032916168, 11.97326256028608982065232159854

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