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Properties

Label 2-325-13.3-c1-0-7
Degree $2$
Conductor $325$
Sign $-0.962 - 0.271i$
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  + (1.27 + 2.20i)2-s + (1.07 + 1.86i)3-s + (−2.24 + 3.88i)4-s + (−2.74 + 4.74i)6-s + (1.46 − 2.54i)7-s − 6.31·8-s + (−0.817 + 1.41i)9-s + (0.317 + 0.550i)11-s − 9.64·12-s + (0.0716 − 3.60i)13-s + 7.48·14-s + (−3.55 − 6.16i)16-s + (0.611 − 1.05i)17-s − 4.16·18-s + (−0.682 + 1.18i)19-s + ⋯
L(s)  = 1  + (0.900 + 1.55i)2-s + (0.621 + 1.07i)3-s + (−1.12 + 1.94i)4-s + (−1.11 + 1.93i)6-s + (0.555 − 0.961i)7-s − 2.23·8-s + (−0.272 + 0.472i)9-s + (0.0957 + 0.165i)11-s − 2.78·12-s + (0.0198 − 0.999i)13-s + 1.99·14-s + (−0.889 − 1.54i)16-s + (0.148 − 0.257i)17-s − 0.981·18-s + (−0.156 + 0.271i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.962 - 0.271i$
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: \((2,\ 325,\ (\ :1/2),\ -0.962 - 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327127 + 2.36177i\)
\(L(\frac12)\) \(\approx\) \(0.327127 + 2.36177i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0716 + 3.60i)T \)
good2 \( 1 + (-1.27 - 2.20i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.07 - 1.86i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.46 + 2.54i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.317 - 0.550i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.611 + 1.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.682 - 1.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.07 + 1.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.96T + 31T^{2} \)
37 \( 1 + (0.611 + 1.05i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.98 - 8.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.683 + 1.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.16T + 47T^{2} \)
53 \( 1 - 0.642T + 53T^{2} \)
59 \( 1 + (3.79 - 6.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.13 + 1.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.01 + 6.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.31 - 2.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 - 1.03T + 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + (-6.27 - 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.39 + 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.45794148892052739408880170058, −10.97527666464036421200514756669, −10.00367180618651699751745532524, −8.937047650385393006247904905973, −7.964010135891601897009369563504, −7.34207561231148335745693681702, −6.05085049776910989552099171525, −4.92998974464495646556625017487, −4.18439920207263686643625445118, −3.31558468809429596259582130457, 1.64041750570228439160641096303, 2.29052616586120195996422410584, 3.57770421711716431169448419990, 4.89646217347186906697171375936, 5.97369905779969673721732396668, 7.35054042969334809019636450952, 8.674085187635423341269318486757, 9.321211613282834939843608277323, 10.66854504508183601540300850491, 11.51819977464117241737717010350

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