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Properties

Label 2-325-13.11-c2-0-35
Degree $2$
Conductor $325$
Sign $-0.844 - 0.535i$
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  + (0.948 − 3.54i)2-s + (1.15 + 1.99i)3-s + (−8.17 − 4.71i)4-s + (8.16 − 2.18i)6-s + (−0.462 − 1.72i)7-s + (−14.0 + 14.0i)8-s + (1.84 − 3.19i)9-s + (−9.34 − 2.50i)11-s − 21.7i·12-s + (−10.6 − 7.38i)13-s − 6.54·14-s + (17.6 + 30.5i)16-s + (−28.7 − 16.6i)17-s + (−9.55 − 9.55i)18-s + (−7.08 + 1.89i)19-s + ⋯
L(s)  = 1  + (0.474 − 1.77i)2-s + (0.384 + 0.665i)3-s + (−2.04 − 1.17i)4-s + (1.36 − 0.364i)6-s + (−0.0660 − 0.246i)7-s + (−1.76 + 1.76i)8-s + (0.204 − 0.354i)9-s + (−0.849 − 0.227i)11-s − 1.81i·12-s + (−0.822 − 0.568i)13-s − 0.467·14-s + (1.10 + 1.91i)16-s + (−1.69 − 0.977i)17-s + (−0.530 − 0.530i)18-s + (−0.372 + 0.0999i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.844 - 0.535i$
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: \((2,\ 325,\ (\ :1),\ -0.844 - 0.535i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.369301 + 1.27090i\)
\(L(\frac12)\) \(\approx\) \(0.369301 + 1.27090i\)
\(L(2)\) 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 \( 1 + (10.6 + 7.38i)T \)
good2 \( 1 + (-0.948 + 3.54i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-1.15 - 1.99i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (0.462 + 1.72i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (9.34 + 2.50i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (28.7 + 16.6i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (7.08 - 1.89i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-30.4 + 17.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-16.4 - 28.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-3.79 - 3.79i)T + 961iT^{2} \)
37 \( 1 + (-20.5 - 5.51i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (1.73 - 6.47i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (46.9 + 27.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-36.1 + 36.1i)T - 2.20e3iT^{2} \)
53 \( 1 + 60.4T + 2.80e3T^{2} \)
59 \( 1 + (-21.4 - 80.2i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-5.88 + 10.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-7.91 + 29.5i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-116. + 31.2i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-61.5 + 61.5i)T - 5.32e3iT^{2} \)
79 \( 1 + 73.3T + 6.24e3T^{2} \)
83 \( 1 + (96.4 + 96.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (10.0 + 2.70i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-161. + 43.3i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−10.66853308663421714467010312351, −10.30618760102779995722980736036, −9.283215140765787438148148110095, −8.634062520772466566521540737052, −6.89945947229677591935748283482, −5.06528298542537568091821140320, −4.51063856557834043833326886558, −3.23669214774358705223085402305, −2.46058447799431322129612283816, −0.49520273077095135439340296569, 2.41894278163605495737199873095, 4.31815075349146330533958878986, 5.10442695377170692903879174005, 6.39126876266766597857238424574, 7.05782782770893948048330432887, 7.911631908111135705639550871131, 8.603111337549763414361712765286, 9.625529936517323984950992236974, 11.08027308299584855930126026041, 12.64147034254839754017920931035

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