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Properties

Label 2-325-65.49-c1-0-15
Degree $2$
Conductor $325$
Sign $0.387 + 0.921i$
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  + (−0.109 − 0.190i)2-s + (1.38 − 0.800i)3-s + (0.975 − 1.69i)4-s + (−0.304 − 0.175i)6-s + (0.166 − 0.287i)7-s − 0.868·8-s + (−0.219 + 0.380i)9-s + (4.65 − 2.68i)11-s − 3.12i·12-s + (−0.619 + 3.55i)13-s − 0.0729·14-s + (−1.85 − 3.21i)16-s + (−4.38 − 2.53i)17-s + 0.0965·18-s + (1.96 + 1.13i)19-s + ⋯
L(s)  = 1  + (−0.0776 − 0.134i)2-s + (0.800 − 0.461i)3-s + (0.487 − 0.845i)4-s + (−0.124 − 0.0717i)6-s + (0.0627 − 0.108i)7-s − 0.306·8-s + (−0.0732 + 0.126i)9-s + (1.40 − 0.809i)11-s − 0.901i·12-s + (−0.171 + 0.985i)13-s − 0.0195·14-s + (−0.464 − 0.803i)16-s + (−1.06 − 0.614i)17-s + 0.0227·18-s + (0.450 + 0.260i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.387 + 0.921i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.387 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48296 - 0.985153i\)
\(L(\frac12)\) \(\approx\) \(1.48296 - 0.985153i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.619 - 3.55i)T \)
good2 \( 1 + (0.109 + 0.190i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.38 + 0.800i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.166 + 0.287i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.65 + 2.68i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.38 + 2.53i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.96 - 1.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.45 - 1.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.45 + 2.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.46iT - 31T^{2} \)
37 \( 1 + (2.98 + 5.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.23 + 1.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.38 - 2.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.34T + 47T^{2} \)
53 \( 1 - 1.56iT - 53T^{2} \)
59 \( 1 + (2.34 + 1.35i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.05 - 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.16 - 8.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.0 + 6.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.68T + 73T^{2} \)
79 \( 1 + 4.51T + 79T^{2} \)
83 \( 1 + 4.26T + 83T^{2} \)
89 \( 1 + (-2.79 + 1.61i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.25 - 2.17i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.42215901548808853320863104686, −10.63534171571221031470999342856, −9.183630469820131849366363593283, −9.005344378723098978345427260780, −7.49432970771379163297824714727, −6.69485409734771248157323008385, −5.67580379803804475316882952742, −4.14248105441800728234912013051, −2.62271395846107507011782811266, −1.45662431761944924064373889095, 2.26464725742882068080148381166, 3.49297335756398902839334460372, 4.33745019915534141670267124622, 6.13228907027543124448174181299, 7.09308284163895412403006615173, 8.131981452140506125472744993788, 8.918211022065597773670215281743, 9.661291897130417349235909995403, 10.88140621668880494190751476471, 11.90951777989641287716669476862

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