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Properties

Label 2-325-13.9-c1-0-0
Degree $2$
Conductor $325$
Sign $-0.213 + 0.976i$
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  + (−0.547 + 0.949i)2-s + (−1.68 + 2.91i)3-s + (0.399 + 0.691i)4-s + (−1.84 − 3.19i)6-s + (−0.795 − 1.37i)7-s − 3.06·8-s + (−4.16 − 7.20i)9-s + (−1.84 + 3.19i)11-s − 2.68·12-s + (3.21 + 1.63i)13-s + 1.74·14-s + (0.882 − 1.52i)16-s + (−1.49 − 2.58i)17-s + 9.11·18-s + (1.39 + 2.42i)19-s + ⋯
L(s)  = 1  + (−0.387 + 0.671i)2-s + (−0.971 + 1.68i)3-s + (0.199 + 0.345i)4-s + (−0.752 − 1.30i)6-s + (−0.300 − 0.520i)7-s − 1.08·8-s + (−1.38 − 2.40i)9-s + (−0.555 + 0.962i)11-s − 0.775·12-s + (0.891 + 0.453i)13-s + 0.466·14-s + (0.220 − 0.381i)16-s + (−0.361 − 0.626i)17-s + 2.14·18-s + (0.321 + 0.556i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.213 + 0.976i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.213 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259719 - 0.322727i\)
\(L(\frac12)\) \(\approx\) \(0.259719 - 0.322727i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.21 - 1.63i)T \)
good2 \( 1 + (0.547 - 0.949i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.68 - 2.91i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.795 + 1.37i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.84 - 3.19i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.49 + 2.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.39 - 2.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.335 - 0.581i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.37 - 2.38i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.43T + 31T^{2} \)
37 \( 1 + (0.479 - 0.829i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.42 - 2.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.44 - 4.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.83T + 47T^{2} \)
53 \( 1 + 2.70T + 53T^{2} \)
59 \( 1 + (-5.17 - 8.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.71 + 6.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.76 - 9.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.28 + 7.42i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.72T + 73T^{2} \)
79 \( 1 - 4.33T + 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 + (0.826 - 1.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.12 + 1.95i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.91300881827255369169898783041, −11.22287276814048962237666425118, −10.31818802703695770858273103855, −9.522083296482535727136917589320, −8.766788680741436015635418004468, −7.36557956553132273570047350687, −6.40189669876616271690706526688, −5.43099376035393462081097303927, −4.28453488737234209050662763348, −3.30855400539417621871621452516, 0.36304622185617679769022233344, 1.73230534991772285646266140660, 2.92536223700126030693975628002, 5.57294638996977729968995568144, 5.92451738963955669441373586872, 6.90152285643176020934443795974, 8.112937238503238129273530333861, 8.955996774465326121344576338919, 10.54401821650306555100958009300, 11.07301302754863353634735324716

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