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Properties

Label 2-325-65.63-c1-0-4
Degree $2$
Conductor $325$
Sign $0.0511 + 0.998i$
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  + (−1.39 − 2.40i)2-s + (0.196 − 0.732i)3-s + (−2.86 + 4.96i)4-s + (−2.03 + 0.545i)6-s + (1.71 + 0.989i)7-s + 10.3·8-s + (2.10 + 1.21i)9-s + (2.64 + 0.707i)11-s + (3.07 + 3.07i)12-s + (−2.83 + 2.22i)13-s − 5.50i·14-s + (−8.71 − 15.0i)16-s + (4.08 − 1.09i)17-s − 6.74i·18-s + (0.742 + 2.77i)19-s + ⋯
L(s)  = 1  + (−0.983 − 1.70i)2-s + (0.113 − 0.422i)3-s + (−1.43 + 2.48i)4-s + (−0.831 + 0.222i)6-s + (0.647 + 0.373i)7-s + 3.67·8-s + (0.700 + 0.404i)9-s + (0.796 + 0.213i)11-s + (0.887 + 0.887i)12-s + (−0.785 + 0.618i)13-s − 1.47i·14-s + (−2.17 − 3.77i)16-s + (0.990 − 0.265i)17-s − 1.58i·18-s + (0.170 + 0.635i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0511 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0511 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.0511 + 0.998i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.0511 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.641745 - 0.609731i\)
\(L(\frac12)\) \(\approx\) \(0.641745 - 0.609731i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.83 - 2.22i)T \)
good2 \( 1 + (1.39 + 2.40i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.196 + 0.732i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.71 - 0.989i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.64 - 0.707i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-4.08 + 1.09i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.742 - 2.77i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.16 + 0.847i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (0.288 - 0.166i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.77 - 3.77i)T + 31iT^{2} \)
37 \( 1 + (0.539 - 0.311i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.894 + 3.33i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.20 + 8.21i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 5.83iT - 47T^{2} \)
53 \( 1 + (-3.61 - 3.61i)T + 53iT^{2} \)
59 \( 1 + (-7.72 + 2.07i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.34 + 9.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.571 + 0.989i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.43 - 1.72i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 4.34T + 73T^{2} \)
79 \( 1 - 4.93iT - 79T^{2} \)
83 \( 1 + 2.83iT - 83T^{2} \)
89 \( 1 + (-1.09 + 4.07i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.23 - 9.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.52796612192874271063072210709, −10.30547540276424036404384995261, −9.790938734960192822222662672482, −8.767373557599171124119104733184, −7.918674863086983463029824220988, −7.10717544103231316906946652307, −4.88139204638036077269814616885, −3.80845291979285174311311126876, −2.29878464665070753489071501583, −1.39287702040124156312644256313, 1.11851536739257520028565603232, 4.15565069764849856763268504740, 5.11780742202160366006989751196, 6.21619816150662112497276987817, 7.23708446142938220077167872893, 7.917199844986876196884764497408, 8.873575325694976320676997886633, 9.869830598917144261493795104091, 10.20164513595804736686347481347, 11.55612028606805620868605696699

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