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Properties

Label 2-325-13.5-c2-0-37
Degree $2$
Conductor $325$
Sign $-0.986 - 0.162i$
Analytic cond. $8.85560$
Root an. cond. $2.97583$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.88 − 1.88i)2-s − 0.793·3-s − 3.11i·4-s + (−1.49 + 1.49i)6-s + (−8.42 − 8.42i)7-s + (1.66 + 1.66i)8-s − 8.36·9-s + (−1.45 − 1.45i)11-s + 2.47i·12-s + (−11.6 − 5.73i)13-s − 31.7·14-s + 18.7·16-s − 2.73i·17-s + (−15.7 + 15.7i)18-s + (−2.87 + 2.87i)19-s + ⋯
L(s)  = 1  + (0.943 − 0.943i)2-s − 0.264·3-s − 0.779i·4-s + (−0.249 + 0.249i)6-s + (−1.20 − 1.20i)7-s + (0.208 + 0.208i)8-s − 0.929·9-s + (−0.132 − 0.132i)11-s + 0.206i·12-s + (−0.897 − 0.441i)13-s − 2.27·14-s + 1.17·16-s − 0.161i·17-s + (−0.877 + 0.877i)18-s + (−0.151 + 0.151i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.986 - 0.162i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ -0.986 - 0.162i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0967842 + 1.18499i\)
\(L(\frac12)\) \(\approx\) \(0.0967842 + 1.18499i\)
\(L(2)\) 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 \( 1 + (11.6 + 5.73i)T \)
good2 \( 1 + (-1.88 + 1.88i)T - 4iT^{2} \)
3 \( 1 + 0.793T + 9T^{2} \)
7 \( 1 + (8.42 + 8.42i)T + 49iT^{2} \)
11 \( 1 + (1.45 + 1.45i)T + 121iT^{2} \)
17 \( 1 + 2.73iT - 289T^{2} \)
19 \( 1 + (2.87 - 2.87i)T - 361iT^{2} \)
23 \( 1 + 41.2iT - 529T^{2} \)
29 \( 1 + 22.4T + 841T^{2} \)
31 \( 1 + (25.5 - 25.5i)T - 961iT^{2} \)
37 \( 1 + (-22.4 - 22.4i)T + 1.36e3iT^{2} \)
41 \( 1 + (-35.3 + 35.3i)T - 1.68e3iT^{2} \)
43 \( 1 + 4.97iT - 1.84e3T^{2} \)
47 \( 1 + (-16.0 - 16.0i)T + 2.20e3iT^{2} \)
53 \( 1 - 55.8T + 2.80e3T^{2} \)
59 \( 1 + (30.7 + 30.7i)T + 3.48e3iT^{2} \)
61 \( 1 + 4.68T + 3.72e3T^{2} \)
67 \( 1 + (-64.9 + 64.9i)T - 4.48e3iT^{2} \)
71 \( 1 + (-2.47 + 2.47i)T - 5.04e3iT^{2} \)
73 \( 1 + (75.2 + 75.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 67.8T + 6.24e3T^{2} \)
83 \( 1 + (-91.5 + 91.5i)T - 6.88e3iT^{2} \)
89 \( 1 + (45.9 + 45.9i)T + 7.92e3iT^{2} \)
97 \( 1 + (111. - 111. i)T - 9.40e3iT^{2} \)
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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

−10.74908407590962697756502067708, −10.56525756428700658312881851405, −9.356003785956846363242861307479, −7.950557456036791334639839954385, −6.82560920252981432605975765512, −5.68919405266951951334088451352, −4.56168309472380925083391225262, −3.48141668473876199304865885040, −2.57677300800520647744959572947, −0.39747891101339107210610668845, 2.58817742878903638929740705985, 3.88209016347457928270020667277, 5.42982356362358366970093751629, 5.74112064196726479367372124652, 6.76020729833150122323903154070, 7.74173054758824306941676889305, 9.142365588686834183216973055755, 9.758743280462729849266599384567, 11.24976915904310447273854352726, 12.14270387920425715492179104890

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