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Properties

Label 2-325-65.18-c1-0-14
Degree $2$
Conductor $325$
Sign $0.329 + 0.944i$
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  + 2.38i·2-s + (−0.648 − 0.648i)3-s − 3.67·4-s + (1.54 − 1.54i)6-s − 2.38·7-s − 3.99i·8-s − 2.15i·9-s + (−3.88 − 3.88i)11-s + (2.38 + 2.38i)12-s + (−2.76 + 2.31i)13-s − 5.67i·14-s + 2.15·16-s + (−0.262 − 0.262i)17-s + 5.14·18-s + (0.587 + 0.587i)19-s + ⋯
L(s)  = 1  + 1.68i·2-s + (−0.374 − 0.374i)3-s − 1.83·4-s + (0.630 − 0.630i)6-s − 0.900·7-s − 1.41i·8-s − 0.719i·9-s + (−1.17 − 1.17i)11-s + (0.687 + 0.687i)12-s + (−0.767 + 0.640i)13-s − 1.51i·14-s + 0.539·16-s + (−0.0637 − 0.0637i)17-s + 1.21·18-s + (0.134 + 0.134i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.329 + 0.944i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115436 - 0.0820068i\)
\(L(\frac12)\) \(\approx\) \(0.115436 - 0.0820068i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.76 - 2.31i)T \)
good2 \( 1 - 2.38iT - 2T^{2} \)
3 \( 1 + (0.648 + 0.648i)T + 3iT^{2} \)
7 \( 1 + 2.38T + 7T^{2} \)
11 \( 1 + (3.88 + 3.88i)T + 11iT^{2} \)
17 \( 1 + (0.262 + 0.262i)T + 17iT^{2} \)
19 \( 1 + (-0.587 - 0.587i)T + 19iT^{2} \)
23 \( 1 + (-1.98 + 1.98i)T - 23iT^{2} \)
29 \( 1 - 7.92iT - 29T^{2} \)
31 \( 1 + (-6.59 + 6.59i)T - 31iT^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 + (6.37 - 6.37i)T - 41iT^{2} \)
43 \( 1 + (-3.09 + 3.09i)T - 43iT^{2} \)
47 \( 1 + 5.67T + 47T^{2} \)
53 \( 1 + (-1.54 - 1.54i)T + 53iT^{2} \)
59 \( 1 + (5.49 - 5.49i)T - 59iT^{2} \)
61 \( 1 - 4.11T + 61T^{2} \)
67 \( 1 + 3.74iT - 67T^{2} \)
71 \( 1 + (2.04 - 2.04i)T - 71iT^{2} \)
73 \( 1 + 7.56iT - 73T^{2} \)
79 \( 1 + 6.07iT - 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 + (-2.23 + 2.23i)T - 89iT^{2} \)
97 \( 1 - 1.88iT - 97T^{2} \)
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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

−11.57501863217132959448153525706, −10.23753611071187300507301467048, −9.199187286442710626469758685931, −8.397312818247469682996128778516, −7.28720709553042343935368293957, −6.58254372554846991156255147337, −5.83701697191244495828814274897, −4.85681546942760525330045832233, −3.21157683174787472917781739898, −0.099305387926241839635564063081, 2.18823558191786331387893412864, 3.16295269506788161285369252656, 4.59688139032288010922313984845, 5.30058118984371356683309833646, 7.07443022090989052771686610028, 8.297979578988262820331734505200, 9.749647834738016582507203202527, 10.06158704082668208925982626435, 10.69789155009804562587835585040, 11.77345755251639954441316676941

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