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Properties

Label 2-325-65.7-c1-0-0
Degree $2$
Conductor $325$
Sign $0.718 - 0.695i$
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  + (−1.52 − 0.880i)2-s + (−0.845 − 3.15i)3-s + (0.550 + 0.953i)4-s + (−1.48 + 5.55i)6-s + (0.694 + 1.20i)7-s + 1.58i·8-s + (−6.63 + 3.82i)9-s + (0.949 + 3.54i)11-s + (2.54 − 2.54i)12-s + (−1.51 + 3.27i)13-s − 2.44i·14-s + (2.49 − 4.32i)16-s + (−3.92 − 1.05i)17-s + 13.4·18-s + (−2.25 − 0.603i)19-s + ⋯
L(s)  = 1  + (−1.07 − 0.622i)2-s + (−0.487 − 1.82i)3-s + (0.275 + 0.476i)4-s + (−0.607 + 2.26i)6-s + (0.262 + 0.454i)7-s + 0.559i·8-s + (−2.21 + 1.27i)9-s + (0.286 + 1.06i)11-s + (0.734 − 0.734i)12-s + (−0.421 + 0.906i)13-s − 0.654i·14-s + (0.623 − 1.08i)16-s + (−0.952 − 0.255i)17-s + 3.17·18-s + (−0.516 − 0.138i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.718 - 0.695i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.718 - 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.110454 + 0.0446836i\)
\(L(\frac12)\) \(\approx\) \(0.110454 + 0.0446836i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 - 3.27i)T \)
good2 \( 1 + (1.52 + 0.880i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.845 + 3.15i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.694 - 1.20i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.949 - 3.54i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.92 + 1.05i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.25 + 0.603i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.51 - 1.47i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.47 + 1.42i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.220 - 0.220i)T + 31iT^{2} \)
37 \( 1 + (-3.03 + 5.26i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.9 - 2.94i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.43 - 5.35i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 5.06T + 47T^{2} \)
53 \( 1 + (0.586 - 0.586i)T - 53iT^{2} \)
59 \( 1 + (0.878 - 3.27i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.03 + 5.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.14 - 0.660i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.41 + 5.29i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 2.15iT - 73T^{2} \)
79 \( 1 - 3.44iT - 79T^{2} \)
83 \( 1 - 1.38T + 83T^{2} \)
89 \( 1 + (1.09 - 0.293i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-13.7 + 7.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.73452901312425769181295875418, −11.08130070512528717085969894683, −9.792636663171002464392488055219, −8.839072385752060094937214729147, −7.974822207534963725826398808747, −7.11232555862501259782613606692, −6.21493695329179921007004332719, −4.89433961425397990108397135262, −2.24155348567440584300874273095, −1.76319160261284920430889854252, 0.12290938874630810334842436168, 3.44741478340819204125195358239, 4.37916513367744476894683687085, 5.63858840341987483636294627862, 6.58348743955074156704682078340, 8.153566090094771635520473623602, 8.702791588022055127380756882830, 9.674145302351708478459810829076, 10.40571571600142919981654655927, 10.92135435043914751354566865917

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