Properties

Degree 2
Conductor $ 5^{2} $
Sign $0.793 - 0.608i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 1.22i)2-s + (−1.22 + 1.22i)3-s − 1.00i·4-s − 2.99·6-s + (−4.89 − 4.89i)7-s + (6.12 − 6.12i)8-s + 6i·9-s − 3·11-s + (1.22 + 1.22i)12-s + (−7.34 + 7.34i)13-s − 11.9i·14-s + 10.9·16-s + (13.4 + 13.4i)17-s + (−7.34 + 7.34i)18-s + 5i·19-s + ⋯
L(s)  = 1  + (0.612 + 0.612i)2-s + (−0.408 + 0.408i)3-s − 0.250i·4-s − 0.499·6-s + (−0.699 − 0.699i)7-s + (0.765 − 0.765i)8-s + 0.666i·9-s − 0.272·11-s + (0.102 + 0.102i)12-s + (−0.565 + 0.565i)13-s − 0.857i·14-s + 0.687·16-s + (0.792 + 0.792i)17-s + (−0.408 + 0.408i)18-s + 0.263i·19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(25\)    =    \(5^{2}\)
\( \varepsilon \)  =  $0.793 - 0.608i$
motivic weight  =  \(2\)
character  :  $\chi_{25} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 25,\ (\ :1),\ 0.793 - 0.608i)$
$L(\frac{3}{2})$  $\approx$  $1.00672 + 0.341600i$
$L(\frac12)$  $\approx$  $1.00672 + 0.341600i$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 5$, \(F_p\) is a polynomial of degree 2. If $p = 5$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 \)
good2 \( 1 + (-1.22 - 1.22i)T + 4iT^{2} \)
3 \( 1 + (1.22 - 1.22i)T - 9iT^{2} \)
7 \( 1 + (4.89 + 4.89i)T + 49iT^{2} \)
11 \( 1 + 3T + 121T^{2} \)
13 \( 1 + (7.34 - 7.34i)T - 169iT^{2} \)
17 \( 1 + (-13.4 - 13.4i)T + 289iT^{2} \)
19 \( 1 - 5iT - 361T^{2} \)
23 \( 1 + (-17.1 + 17.1i)T - 529iT^{2} \)
29 \( 1 + 30iT - 841T^{2} \)
31 \( 1 + 38T + 961T^{2} \)
37 \( 1 + (-19.5 - 19.5i)T + 1.36e3iT^{2} \)
41 \( 1 - 57T + 1.68e3T^{2} \)
43 \( 1 + (-4.89 + 4.89i)T - 1.84e3iT^{2} \)
47 \( 1 + (-7.34 - 7.34i)T + 2.20e3iT^{2} \)
53 \( 1 + (31.8 - 31.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 90iT - 3.48e3T^{2} \)
61 \( 1 + 28T + 3.72e3T^{2} \)
67 \( 1 + (47.7 + 47.7i)T + 4.48e3iT^{2} \)
71 \( 1 - 42T + 5.04e3T^{2} \)
73 \( 1 + (13.4 - 13.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 80iT - 6.24e3T^{2} \)
83 \( 1 + (111. - 111. i)T - 6.88e3iT^{2} \)
89 \( 1 + 15iT - 7.92e3T^{2} \)
97 \( 1 + (53.8 + 53.8i)T + 9.40e3iT^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.85019171884077818291484826468, −16.37859451314354405342318734091, −15.05651061438866677572858832335, −13.91243975330688633400572398253, −12.77679944275163605300715426036, −10.80202217950719097990488636664, −9.843700278017067399731181305591, −7.43070867433979855000875174944, −5.90391852040235819298130691681, −4.38168152664509159724498301141, 3.11856284495437891957650319790, 5.45386990868911896353903542376, 7.36806039937676854566796043849, 9.354066587400336108683727151781, 11.20975637095062978438764001943, 12.39871966517840062865009927183, 12.92108624713706615057147601345, 14.56750584768152811512563579792, 16.06656954705346165557575563172, 17.37879555976911716170250167333

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