Properties

Label 2-325-5.4-c5-0-63
Degree $2$
Conductor $325$
Sign $-0.447 + 0.894i$
Analytic cond. $52.1247$
Root an. cond. $7.21974$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.78i·2-s − 8.20i·3-s + 17.6·4-s − 31.0·6-s + 88.9i·7-s − 188. i·8-s + 175.·9-s − 156.·11-s − 145. i·12-s − 169i·13-s + 336.·14-s − 145.·16-s + 447. i·17-s − 664. i·18-s + 269.·19-s + ⋯
L(s)  = 1  − 0.668i·2-s − 0.526i·3-s + 0.552·4-s − 0.352·6-s + 0.686i·7-s − 1.03i·8-s + 0.722·9-s − 0.390·11-s − 0.290i·12-s − 0.277i·13-s + 0.459·14-s − 0.142·16-s + 0.375i·17-s − 0.483i·18-s + 0.171·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(52.1247\)
Root analytic conductor: \(7.21974\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :5/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.625025574\)
\(L(\frac12)\) \(\approx\) \(2.625025574\)
\(L(\frac{7}{2})\) 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 \( 1 + 169iT \)
good2 \( 1 + 3.78iT - 32T^{2} \)
3 \( 1 + 8.20iT - 243T^{2} \)
7 \( 1 - 88.9iT - 1.68e4T^{2} \)
11 \( 1 + 156.T + 1.61e5T^{2} \)
17 \( 1 - 447. iT - 1.41e6T^{2} \)
19 \( 1 - 269.T + 2.47e6T^{2} \)
23 \( 1 + 1.37e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.69e3T + 2.05e7T^{2} \)
31 \( 1 - 797.T + 2.86e7T^{2} \)
37 \( 1 + 4.39e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.43e4T + 1.15e8T^{2} \)
43 \( 1 + 1.13e4iT - 1.47e8T^{2} \)
47 \( 1 + 9.97e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.15e4iT - 4.18e8T^{2} \)
59 \( 1 + 7.13e3T + 7.14e8T^{2} \)
61 \( 1 - 1.66e4T + 8.44e8T^{2} \)
67 \( 1 - 4.21e3iT - 1.35e9T^{2} \)
71 \( 1 - 1.28e4T + 1.80e9T^{2} \)
73 \( 1 - 1.30e4iT - 2.07e9T^{2} \)
79 \( 1 + 7.47e4T + 3.07e9T^{2} \)
83 \( 1 + 8.54e3iT - 3.93e9T^{2} \)
89 \( 1 + 5.95e4T + 5.58e9T^{2} \)
97 \( 1 + 1.73e4iT - 8.58e9T^{2} \)
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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

−10.52170641979929578500788896811, −9.833681550909324982747309189538, −8.583251504434373456831021302036, −7.51285482202065913064276508399, −6.66448648677402000510918547861, −5.64230680119232670955078797939, −4.14834568141101641153408609077, −2.79319066634405293823635598369, −1.92290755612145368656059573563, −0.72195375650107180667651601060, 1.24971763256407764072469099259, 2.77706767818221027446723659999, 4.17315358991876740513198613666, 5.15543729445058983672468479508, 6.35760708581194387992497164088, 7.26330665869461945153923952714, 7.932439714749123117678739107948, 9.244983650484167423809243783536, 10.21282391674200342950759019979, 10.92478876659734288405772046629

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