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Properties

Label 2-325-5.4-c5-0-14
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  + 7.28i·2-s + 14.9i·3-s − 21.0·4-s − 109.·6-s + 135. i·7-s + 79.9i·8-s + 18.1·9-s + 191.·11-s − 315. i·12-s − 169i·13-s − 985.·14-s − 1.25e3·16-s − 874. i·17-s + 132. i·18-s − 1.99e3·19-s + ⋯
L(s)  = 1  + 1.28i·2-s + 0.961i·3-s − 0.656·4-s − 1.23·6-s + 1.04i·7-s + 0.441i·8-s + 0.0748·9-s + 0.476·11-s − 0.631i·12-s − 0.277i·13-s − 1.34·14-s − 1.22·16-s − 0.733i·17-s + 0.0963i·18-s − 1.26·19-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

\(L(3)\) \(\approx\) \(1.115833971\)
\(L(\frac12)\) \(\approx\) \(1.115833971\)
\(L(\frac{7}{2})\) 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 \( 1 - 7.28iT - 32T^{2} \)
3 \( 1 - 14.9iT - 243T^{2} \)
7 \( 1 - 135. iT - 1.68e4T^{2} \)
11 \( 1 - 191.T + 1.61e5T^{2} \)
17 \( 1 + 874. iT - 1.41e6T^{2} \)
19 \( 1 + 1.99e3T + 2.47e6T^{2} \)
23 \( 1 - 2.09e3iT - 6.43e6T^{2} \)
29 \( 1 - 46.3T + 2.05e7T^{2} \)
31 \( 1 + 9.45e3T + 2.86e7T^{2} \)
37 \( 1 + 3.37e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.05e4T + 1.15e8T^{2} \)
43 \( 1 + 4.05e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.70e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.39e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.01e4T + 7.14e8T^{2} \)
61 \( 1 + 2.74e4T + 8.44e8T^{2} \)
67 \( 1 + 317. iT - 1.35e9T^{2} \)
71 \( 1 - 3.96e4T + 1.80e9T^{2} \)
73 \( 1 + 4.76e3iT - 2.07e9T^{2} \)
79 \( 1 - 7.45e4T + 3.07e9T^{2} \)
83 \( 1 - 1.42e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.32e5T + 5.58e9T^{2} \)
97 \( 1 - 1.93e4iT - 8.58e9T^{2} \)
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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

−11.42758280625037475533058061364, −10.50152883012026997341774828924, −9.232717286880752918463405580491, −8.873021483246194207374195342656, −7.65650558381964430119226894127, −6.67536926932661694379475342111, −5.62020375057735067447250347273, −4.94839816640159478449104223288, −3.70176720004124815817578196677, −2.11426362595565290552993169300, 0.28018021364497598991012094497, 1.37378004706762973458519324724, 2.11760821420929738348589662071, 3.63776908646797401063906132089, 4.42764840796111830302177476609, 6.41623335791145311470242217024, 6.98024535387071298113436536491, 8.117229863790768616422167241696, 9.308899570519870320329400091452, 10.39187597998844865574326374157

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