Properties

Degree 2
Conductor $ 5^{2} $
Sign $0.437 + 0.899i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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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

\[\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(25\)    =    \(5^{2}\)
\( \varepsilon \)  =  $0.437 + 0.899i$
motivic weight  =  \(2\)
character  :  $\chi_{25} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 25,\ (\ :1),\ 0.437 + 0.899i)$
$L(\frac{3}{2})$  $\approx$  $0.672853 - 0.420861i$
$L(\frac12)$  $\approx$  $0.672853 - 0.420861i$
$L(2)$   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

−17.80124358464963284926715165367, −15.87327121129670108706619144263, −14.61987166921431862955652160781, −13.38291394650940071904926706610, −11.72130267965928507532906959660, −10.65256669380755182035270122951, −9.075493128417653782841684094869, −7.902577928817560403427339720800, −5.50779105914296339937998810838, −2.18276126975121981967295505990, 3.96617731422108493669297013406, 6.66659017825642368793563998673, 8.181050810439558547923718676808, 9.231979411563966906372676961873, 10.87036483367697919382493644174, 12.57910233808340025768232932523, 14.17692961093489109108404805037, 15.35474335332985474713232506729, 16.42661081484645235263474578445, 17.56119575775530289336883258965

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