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Properties

Label 2-325-65.18-c1-0-8
Degree $2$
Conductor $325$
Sign $0.845 + 0.533i$
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  − 1.68i·2-s + (1.99 + 1.99i)3-s − 0.844·4-s + (3.36 − 3.36i)6-s + 1.68·7-s − 1.94i·8-s + 4.97i·9-s + (−1.38 − 1.38i)11-s + (−1.68 − 1.68i)12-s + (−0.723 + 3.53i)13-s − 2.84i·14-s − 4.97·16-s + (4.40 + 4.40i)17-s + 8.39·18-s + (−5.89 − 5.89i)19-s + ⋯
L(s)  = 1  − 1.19i·2-s + (1.15 + 1.15i)3-s − 0.422·4-s + (1.37 − 1.37i)6-s + 0.637·7-s − 0.688i·8-s + 1.65i·9-s + (−0.418 − 0.418i)11-s + (−0.486 − 0.486i)12-s + (−0.200 + 0.979i)13-s − 0.760i·14-s − 1.24·16-s + (1.06 + 1.06i)17-s + 1.97·18-s + (−1.35 − 1.35i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.845 + 0.533i$
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: \((2,\ 325,\ (\ :1/2),\ 0.845 + 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92795 - 0.556924i\)
\(L(\frac12)\) \(\approx\) \(1.92795 - 0.556924i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.723 - 3.53i)T \)
good2 \( 1 + 1.68iT - 2T^{2} \)
3 \( 1 + (-1.99 - 1.99i)T + 3iT^{2} \)
7 \( 1 - 1.68T + 7T^{2} \)
11 \( 1 + (1.38 + 1.38i)T + 11iT^{2} \)
17 \( 1 + (-4.40 - 4.40i)T + 17iT^{2} \)
19 \( 1 + (5.89 + 5.89i)T + 19iT^{2} \)
23 \( 1 + (-3.80 + 3.80i)T - 23iT^{2} \)
29 \( 1 - 1.60iT - 29T^{2} \)
31 \( 1 + (1.22 - 1.22i)T - 31iT^{2} \)
37 \( 1 + 7.86T + 37T^{2} \)
41 \( 1 + (-1.76 + 1.76i)T - 41iT^{2} \)
43 \( 1 + (-0.452 + 0.452i)T - 43iT^{2} \)
47 \( 1 - 0.422T + 47T^{2} \)
53 \( 1 + (9.85 + 9.85i)T + 53iT^{2} \)
59 \( 1 + (-0.149 + 0.149i)T - 59iT^{2} \)
61 \( 1 + 3.87T + 61T^{2} \)
67 \( 1 - 11.6iT - 67T^{2} \)
71 \( 1 + (-6.25 + 6.25i)T - 71iT^{2} \)
73 \( 1 - 9.09iT - 73T^{2} \)
79 \( 1 - 4.71iT - 79T^{2} \)
83 \( 1 - 4.89T + 83T^{2} \)
89 \( 1 + (4.59 - 4.59i)T - 89iT^{2} \)
97 \( 1 - 1.47iT - 97T^{2} \)
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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

−11.07864576075706846554878780053, −10.71942776583545697067215622080, −9.820112135883549457070607920120, −8.911261373421142436079033388653, −8.289787688554834758820322990034, −6.80172634362421810152730669094, −4.91301913152355647673760987801, −4.03754319324910202803708663107, −3.04193889551120126544449942713, −1.97099282535672410561531210754, 1.81841438596914432049970808482, 3.10029852348220876006390782490, 5.01038691497593911921607348002, 6.08303843435816747202228783319, 7.26114585726222186541028010110, 7.79148086814216173300159592093, 8.267014196223663310688805645507, 9.378311254910302904044577680164, 10.71733598420532530762583753424, 12.05256622107606308333791291379

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