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Properties

Label 2-325-65.3-c0-0-0
Degree $2$
Conductor $325$
Sign $0.468 - 0.883i$
Analytic cond. $0.162196$
Root an. cond. $0.402735$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.258 + 0.965i)3-s + (−0.499 + 0.866i)6-s + (0.258 + 0.965i)7-s + (−0.707 − 0.707i)8-s + (−0.5 − 0.866i)11-s + (0.707 + 0.707i)13-s + i·14-s + (−0.5 − 0.866i)16-s + (−0.258 − 0.965i)17-s + (−0.866 − 0.5i)19-s − 21-s + (−0.258 − 0.965i)22-s + (0.258 − 0.965i)23-s + (0.866 − 0.499i)24-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.258 + 0.965i)3-s + (−0.499 + 0.866i)6-s + (0.258 + 0.965i)7-s + (−0.707 − 0.707i)8-s + (−0.5 − 0.866i)11-s + (0.707 + 0.707i)13-s + i·14-s + (−0.5 − 0.866i)16-s + (−0.258 − 0.965i)17-s + (−0.866 − 0.5i)19-s − 21-s + (−0.258 − 0.965i)22-s + (0.258 − 0.965i)23-s + (0.866 − 0.499i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.468 - 0.883i$
Analytic conductor: \(0.162196\)
Root analytic conductor: \(0.402735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :0),\ 0.468 - 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.063328779\)
\(L(\frac12)\) \(\approx\) \(1.063328779\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
3 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
7 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
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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

−11.97745328251240419860787322072, −11.17716742363198603676242366172, −10.23426874134615930468138319292, −9.128359529591169715367184205955, −8.536078590989055215976771831270, −6.74392235497705062067962529038, −5.78941786480063852429820169715, −4.92200513007803894603375605824, −4.20004412422117194119859308834, −2.81826998948620157783634034250, 1.77247027103046769908399431599, 3.52621030662695976480044763924, 4.49912473471969393278016859786, 5.69785316935324921116678442293, 6.73378786824217088889563621048, 7.72360873780605816117183590160, 8.590562383179906133587738142728, 10.17083871556584762314021753885, 10.93301705732237245984978849436, 12.08997808522349553622929070982

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