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Properties

Label 2-325-13.11-c2-0-13
Degree $2$
Conductor $325$
Sign $-0.188 - 0.981i$
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  + (−0.106 + 0.396i)2-s + (1.67 + 2.89i)3-s + (3.31 + 1.91i)4-s + (−1.32 + 0.354i)6-s + (−0.223 − 0.834i)7-s + (−2.27 + 2.27i)8-s + (−1.08 + 1.88i)9-s + (7.17 + 1.92i)11-s + 12.8i·12-s + (1.96 + 12.8i)13-s + 0.354·14-s + (7.00 + 12.1i)16-s + (−2.20 − 1.27i)17-s + (−0.632 − 0.632i)18-s + (12.8 − 3.43i)19-s + ⋯
L(s)  = 1  + (−0.0530 + 0.198i)2-s + (0.557 + 0.965i)3-s + (0.829 + 0.478i)4-s + (−0.220 + 0.0591i)6-s + (−0.0319 − 0.119i)7-s + (−0.283 + 0.283i)8-s + (−0.121 + 0.209i)9-s + (0.652 + 0.174i)11-s + 1.06i·12-s + (0.150 + 0.988i)13-s + 0.0252·14-s + (0.437 + 0.758i)16-s + (−0.129 − 0.0748i)17-s + (−0.0351 − 0.0351i)18-s + (0.675 − 0.180i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.188 - 0.981i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ -0.188 - 0.981i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.49393 + 1.80871i\)
\(L(\frac12)\) \(\approx\) \(1.49393 + 1.80871i\)
\(L(2)\) 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 \( 1 + (-1.96 - 12.8i)T \)
good2 \( 1 + (0.106 - 0.396i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-1.67 - 2.89i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (0.223 + 0.834i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-7.17 - 1.92i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (2.20 + 1.27i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-12.8 + 3.43i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (26.5 - 15.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (11.5 + 19.9i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (30.9 + 30.9i)T + 961iT^{2} \)
37 \( 1 + (32.1 + 8.61i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-3.57 + 13.3i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-49.0 - 28.3i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-19.6 + 19.6i)T - 2.20e3iT^{2} \)
53 \( 1 + 41.0T + 2.80e3T^{2} \)
59 \( 1 + (3.49 + 13.0i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (44.1 - 76.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.49 - 9.29i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-121. + 32.6i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-86.4 + 86.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 29.8T + 6.24e3T^{2} \)
83 \( 1 + (106. + 106. i)T + 6.88e3iT^{2} \)
89 \( 1 + (-78.1 - 20.9i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-116. + 31.2i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.61763473241622166944062253942, −10.72314189084329581738379414190, −9.569563364136400701548014911096, −9.048062380084038127561507914336, −7.83137690400938571388941548378, −6.93582329085402733665150298788, −5.84690431417561624108014973277, −4.21512183986886516189294179927, −3.52174954745619756180819443946, −2.04116034368660540783764646345, 1.14480669666583025053419364614, 2.26938593002479733114667816834, 3.45612934267250405751700584929, 5.39171149948673600791249306206, 6.42223247142879304474651816575, 7.28233111729587600125405264083, 8.124467521501019324037682499880, 9.213966224979917905066089812132, 10.36711711873148390974319360910, 11.08369310543307940847892895116

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