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Properties

Label 2-325-65.47-c1-0-12
Degree $2$
Conductor $325$
Sign $0.840 + 0.541i$
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  − 0.438i·2-s + (0.242 − 0.242i)3-s + 1.80·4-s + (−0.106 − 0.106i)6-s − 0.438·7-s − 1.67i·8-s + 2.88i·9-s + (3.04 − 3.04i)11-s + (0.438 − 0.438i)12-s + (2.89 − 2.14i)13-s + 0.192i·14-s + 2.88·16-s + (−3.09 + 3.09i)17-s + 1.26·18-s + (−1.59 + 1.59i)19-s + ⋯
L(s)  = 1  − 0.310i·2-s + (0.140 − 0.140i)3-s + 0.903·4-s + (−0.0434 − 0.0434i)6-s − 0.165·7-s − 0.590i·8-s + 0.960i·9-s + (0.918 − 0.918i)11-s + (0.126 − 0.126i)12-s + (0.802 − 0.596i)13-s + 0.0514i·14-s + 0.720·16-s + (−0.749 + 0.749i)17-s + 0.298·18-s + (−0.365 + 0.365i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.840 + 0.541i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.840 + 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63973 - 0.482210i\)
\(L(\frac12)\) \(\approx\) \(1.63973 - 0.482210i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.89 + 2.14i)T \)
good2 \( 1 + 0.438iT - 2T^{2} \)
3 \( 1 + (-0.242 + 0.242i)T - 3iT^{2} \)
7 \( 1 + 0.438T + 7T^{2} \)
11 \( 1 + (-3.04 + 3.04i)T - 11iT^{2} \)
17 \( 1 + (3.09 - 3.09i)T - 17iT^{2} \)
19 \( 1 + (1.59 - 1.59i)T - 19iT^{2} \)
23 \( 1 + (2.95 + 2.95i)T + 23iT^{2} \)
29 \( 1 - 0.138iT - 29T^{2} \)
31 \( 1 + (1.27 + 1.27i)T + 31iT^{2} \)
37 \( 1 - 7.13T + 37T^{2} \)
41 \( 1 + (-0.0318 - 0.0318i)T + 41iT^{2} \)
43 \( 1 + (-2.20 - 2.20i)T + 43iT^{2} \)
47 \( 1 + 7.44T + 47T^{2} \)
53 \( 1 + (7.84 - 7.84i)T - 53iT^{2} \)
59 \( 1 + (6.01 + 6.01i)T + 59iT^{2} \)
61 \( 1 + 9.35T + 61T^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 + (1.51 + 1.51i)T + 71iT^{2} \)
73 \( 1 - 15.1iT - 73T^{2} \)
79 \( 1 - 10.0iT - 79T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 + (9.98 + 9.98i)T + 89iT^{2} \)
97 \( 1 - 1.50iT - 97T^{2} \)
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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

−11.11033665515820392019154172380, −11.02287033760558092029067927853, −9.863481208747882908923851334996, −8.531871405424272614546735816661, −7.85010596404114037372561073074, −6.49193061322114499145152700585, −5.92673377747614616632330280601, −4.14591867113818077898962533664, −2.94097917101120355379947589392, −1.58056419918068654387705830581, 1.79146594891864313602921899282, 3.35123574419419903330176905049, 4.57107792946006656080760053995, 6.27180061416209934502079712293, 6.63107626900907924473484620194, 7.72729690401024760470718383753, 9.038805420666819366728415781972, 9.624919423411694699020556304569, 10.94749553772391934128983733452, 11.68657397910534009660824620300

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