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Properties

Label 2-325-65.18-c1-0-4
Degree $2$
Conductor $325$
Sign $0.966 - 0.256i$
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  + i·2-s + (−1 − i)3-s + 4-s + (1 − i)6-s + 2·7-s + 3i·8-s − i·9-s + (−1 − i)11-s + (−1 − i)12-s + (2 − 3i)13-s + 2i·14-s − 16-s + (1 + i)17-s + 18-s + (5 + 5i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.577 − 0.577i)3-s + 0.5·4-s + (0.408 − 0.408i)6-s + 0.755·7-s + 1.06i·8-s − 0.333i·9-s + (−0.301 − 0.301i)11-s + (−0.288 − 0.288i)12-s + (0.554 − 0.832i)13-s + 0.534i·14-s − 0.250·16-s + (0.242 + 0.242i)17-s + 0.235·18-s + (1.14 + 1.14i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.966 - 0.256i$
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: \((2,\ 325,\ (\ :1/2),\ 0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39869 + 0.182557i\)
\(L(\frac12)\) \(\approx\) \(1.39869 + 0.182557i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 3i)T \)
good2 \( 1 - iT - 2T^{2} \)
3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + (1 + i)T + 11iT^{2} \)
17 \( 1 + (-1 - i)T + 17iT^{2} \)
19 \( 1 + (-5 - 5i)T + 19iT^{2} \)
23 \( 1 + (-3 + 3i)T - 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-5 + 5i)T - 31iT^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (7 - 7i)T - 41iT^{2} \)
43 \( 1 + (1 - i)T - 43iT^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (5 + 5i)T + 53iT^{2} \)
59 \( 1 + (-7 + 7i)T - 59iT^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + (-1 + i)T - 71iT^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (5 - 5i)T - 89iT^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−11.59263846417408964018128818610, −11.01293290014164402462558776367, −9.873443212325966159410298061836, −8.253305675688649284095761861164, −7.87155574364968762775767305140, −6.68892470576552616216444542324, −5.92458878441165171090142572011, −5.10467461181155124744900307917, −3.20387542892187851939078231485, −1.38698629632226783143308340965, 1.56913327172388475004175740625, 3.05605240781393840939190521912, 4.50679979538928923128421638146, 5.35802131020787838166673963893, 6.75376157360806958794585756715, 7.67109668290368276829886412309, 9.050008011406213315356624561879, 10.05127028613271994248561309330, 10.82669726838418579362701549665, 11.49877995026866319420390841576

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