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Properties

Label 2-325-13.10-c1-0-4
Degree $2$
Conductor $325$
Sign $0.879 + 0.475i$
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  + (−2.27 + 1.31i)2-s + (−1.34 − 2.33i)3-s + (2.46 − 4.26i)4-s + (6.15 + 3.55i)6-s + (2.37 + 1.37i)7-s + 7.71i·8-s + (−2.14 + 3.71i)9-s + (−1.59 + 0.920i)11-s − 13.3·12-s + (3.60 − 0.149i)13-s − 7.21·14-s + (−5.22 − 9.04i)16-s + (1.90 − 3.30i)17-s − 11.2i·18-s + (4.91 + 2.83i)19-s + ⋯
L(s)  = 1  + (−1.61 + 0.930i)2-s + (−0.779 − 1.34i)3-s + (1.23 − 2.13i)4-s + (2.51 + 1.45i)6-s + (0.897 + 0.517i)7-s + 2.72i·8-s + (−0.714 + 1.23i)9-s + (−0.480 + 0.277i)11-s − 3.84·12-s + (0.999 − 0.0413i)13-s − 1.92·14-s + (−1.30 − 2.26i)16-s + (0.462 − 0.801i)17-s − 2.65i·18-s + (1.12 + 0.650i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.505495 - 0.127749i\)
\(L(\frac12)\) \(\approx\) \(0.505495 - 0.127749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 + 0.149i)T \)
good2 \( 1 + (2.27 - 1.31i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.34 + 2.33i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.37 - 1.37i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.59 - 0.920i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.90 + 3.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.91 - 2.83i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0696 - 0.120i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.583 + 1.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.69iT - 31T^{2} \)
37 \( 1 + (2.13 - 1.23i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.01 + 4.62i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.04 + 1.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.66iT - 47T^{2} \)
53 \( 1 - 9.97T + 53T^{2} \)
59 \( 1 + (-2.19 - 1.26i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.17 + 7.23i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.18 - 2.99i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.58 + 4.38i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.7iT - 73T^{2} \)
79 \( 1 + 9.64T + 79T^{2} \)
83 \( 1 - 15.6iT - 83T^{2} \)
89 \( 1 + (-2.91 + 1.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.271 + 0.156i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.46040952772508620205288021634, −10.56764927958175674858630721760, −9.435746629563463881897682649031, −8.348261759870014415599140207219, −7.68867671008725609664663020162, −7.06646642989649457390623476410, −5.88182492707869223154082421535, −5.41322315739245583361023371886, −2.03218642813927162949855548015, −0.891338574915642180377255801781, 1.16503399618265837951525959172, 3.19134646369814284471183164902, 4.30774605135961599612817915899, 5.68389851444403484286420587646, 7.30426153577124667780973840058, 8.319954887254352823491074065487, 9.118521841868029347762798375360, 10.10621659572946434930946703898, 10.67062290496861099560707228835, 11.21029651088173558919520141282

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