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Properties

Label 2-325-65.4-c1-0-5
Degree $2$
Conductor $325$
Sign $0.970 - 0.242i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24982 + 0.154157i\)
\(L(\frac12)\) \(\approx\) \(1.24982 + 0.154157i\)
\(L(\frac{3}{2})\) 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

−11.59887192740928478018404984038, −11.32832787483400199471480157841, −9.730682880579857412391562909001, −8.921962764060130718047141059559, −7.42593704848573239658088243324, −6.92712292154064381025052579688, −5.97768155949823842039786203106, −4.46739520995589349411543927133, −3.31020519528522223993293191442, −1.53917094639609003356313042075, 1.18383005486174056076030179925, 3.25813699324565033418423577693, 4.79602749756193513797099214650, 5.83414922367745099687517492507, 6.24016928428768390060901777130, 7.65157687923208992188878808362, 8.884392418307982028002155721750, 10.06362025872256443315217389780, 10.61385526065486544348156985598, 11.48847235087699649548005123715

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