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Properties

Label 2-325-65.2-c1-0-10
Degree $2$
Conductor $325$
Sign $0.513 + 0.857i$
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  + (−0.880 − 1.52i)2-s + (3.15 + 0.845i)3-s + (−0.550 + 0.953i)4-s + (−1.48 − 5.55i)6-s + (1.20 + 0.694i)7-s − 1.58·8-s + (6.63 + 3.82i)9-s + (0.949 − 3.54i)11-s + (−2.54 + 2.54i)12-s + (−3.27 + 1.51i)13-s − 2.44i·14-s + (2.49 + 4.32i)16-s + (−1.05 − 3.92i)17-s − 13.4i·18-s + (2.25 − 0.603i)19-s + ⋯
L(s)  = 1  + (−0.622 − 1.07i)2-s + (1.82 + 0.487i)3-s + (−0.275 + 0.476i)4-s + (−0.607 − 2.26i)6-s + (0.454 + 0.262i)7-s − 0.559·8-s + (2.21 + 1.27i)9-s + (0.286 − 1.06i)11-s + (−0.734 + 0.734i)12-s + (−0.906 + 0.421i)13-s − 0.654i·14-s + (0.623 + 1.08i)16-s + (−0.255 − 0.952i)17-s − 3.17i·18-s + (0.516 − 0.138i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.513 + 0.857i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.513 + 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51396 - 0.857936i\)
\(L(\frac12)\) \(\approx\) \(1.51396 - 0.857936i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.27 - 1.51i)T \)
good2 \( 1 + (0.880 + 1.52i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-3.15 - 0.845i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.20 - 0.694i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.949 + 3.54i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.05 + 3.92i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.25 + 0.603i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.47 - 5.51i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.47 + 1.42i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.220 + 0.220i)T - 31iT^{2} \)
37 \( 1 + (5.26 - 3.03i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.9 + 2.94i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.35 - 1.43i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 5.06iT - 47T^{2} \)
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 \( 1 + (-0.660 - 1.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.41 - 5.29i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 2.15T + 73T^{2} \)
79 \( 1 - 3.44iT - 79T^{2} \)
83 \( 1 + 1.38iT - 83T^{2} \)
89 \( 1 + (-1.09 - 0.293i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.92 - 13.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.39553841971061193437619242914, −10.16710197107470120739959650716, −9.621336765687710833503493813161, −8.823588230892889306024392160134, −8.252920423489726323835362309974, −7.05623368781569018153992362578, −5.09788271372026659950739211909, −3.63227540692981130687610486526, −2.81903240624668949630119519828, −1.74412553548040956937166677145, 1.90574312872459311911695104568, 3.25321187297253968618511180671, 4.67700063022674278882575111253, 6.56870908942623917678123910868, 7.25562018683874035644766712499, 8.017610453771215549630668948844, 8.562034489857320636033117683142, 9.532278918846201513311816387205, 10.23027077975268206574614633007, 12.19872528140699668030997890884

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