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Properties

Label 2-325-5.4-c5-0-15
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  + 9.27i·2-s + 5.85i·3-s − 54.1·4-s − 54.3·6-s − 90.4i·7-s − 205. i·8-s + 208.·9-s + 597.·11-s − 316. i·12-s − 169i·13-s + 838.·14-s + 172.·16-s + 1.34e3i·17-s + 1.93e3i·18-s − 2.98e3·19-s + ⋯
L(s)  = 1  + 1.64i·2-s + 0.375i·3-s − 1.69·4-s − 0.616·6-s − 0.697i·7-s − 1.13i·8-s + 0.858·9-s + 1.48·11-s − 0.635i·12-s − 0.277i·13-s + 1.14·14-s + 0.168·16-s + 1.13i·17-s + 1.40i·18-s − 1.89·19-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

\(L(3)\) \(\approx\) \(1.256524964\)
\(L(\frac12)\) \(\approx\) \(1.256524964\)
\(L(\frac{7}{2})\) 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 \( 1 - 9.27iT - 32T^{2} \)
3 \( 1 - 5.85iT - 243T^{2} \)
7 \( 1 + 90.4iT - 1.68e4T^{2} \)
11 \( 1 - 597.T + 1.61e5T^{2} \)
17 \( 1 - 1.34e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.98e3T + 2.47e6T^{2} \)
23 \( 1 - 1.35e3iT - 6.43e6T^{2} \)
29 \( 1 + 6.78e3T + 2.05e7T^{2} \)
31 \( 1 - 661.T + 2.86e7T^{2} \)
37 \( 1 - 9.68e3iT - 6.93e7T^{2} \)
41 \( 1 - 6.93e3T + 1.15e8T^{2} \)
43 \( 1 - 2.51e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.02e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.06e3iT - 4.18e8T^{2} \)
59 \( 1 - 7.02e3T + 7.14e8T^{2} \)
61 \( 1 + 3.12e4T + 8.44e8T^{2} \)
67 \( 1 - 2.78e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.90e4T + 1.80e9T^{2} \)
73 \( 1 - 6.90e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.98e4T + 3.07e9T^{2} \)
83 \( 1 - 3.54e4iT - 3.93e9T^{2} \)
89 \( 1 + 7.91e4T + 5.58e9T^{2} \)
97 \( 1 - 5.92e4iT - 8.58e9T^{2} \)
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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

−11.21927522070501832182885933930, −10.22239468723216846865215957894, −9.253714814705280606041133038009, −8.423284365032894678026106216228, −7.39310387972771777917244494574, −6.62780367512104844096366159538, −5.84751013555630590043114248937, −4.34563625381918949397330658723, −3.98831387032827251861588781562, −1.45403863593380759024457359846, 0.33546296109470443883672761727, 1.62243781023993130175905777159, 2.35100940510294265357330748512, 3.81229781654973744883888249089, 4.55585699319447266713161820317, 6.20203936037960732801638984267, 7.24367685040386872336981404468, 8.887723291467197565099059070505, 9.238248359584698173594972129510, 10.31896644427148762774946748097

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