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Properties

Label 2-5e2-5.2-c4-0-4
Degree $2$
Conductor $25$
Sign $-0.793 + 0.608i$
Analytic cond. $2.58424$
Root an. cond. $1.60755$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.67 − 3.67i)2-s + (8.57 − 8.57i)3-s + 11i·4-s − 63.0·6-s + (−34.2 − 34.2i)7-s + (−18.3 + 18.3i)8-s − 66.0i·9-s + 117·11-s + (94.3 + 94.3i)12-s + (51.4 − 51.4i)13-s + 252. i·14-s + 311·16-s + (180. + 180. i)17-s + (−242. + 242. i)18-s − 595i·19-s + ⋯
L(s)  = 1  + (−0.918 − 0.918i)2-s + (0.952 − 0.952i)3-s + 0.687i·4-s − 1.75·6-s + (−0.699 − 0.699i)7-s + (−0.287 + 0.287i)8-s − 0.814i·9-s + 0.966·11-s + (0.654 + 0.654i)12-s + (0.304 − 0.304i)13-s + 1.28i·14-s + 1.21·16-s + (0.622 + 0.622i)17-s + (−0.748 + 0.748i)18-s − 1.64i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.793 + 0.608i$
Analytic conductor: \(2.58424\)
Root analytic conductor: \(1.60755\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :2),\ -0.793 + 0.608i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.333380 - 0.982502i\)
\(L(\frac12)\) \(\approx\) \(0.333380 - 0.982502i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (3.67 + 3.67i)T + 16iT^{2} \)
3 \( 1 + (-8.57 + 8.57i)T - 81iT^{2} \)
7 \( 1 + (34.2 + 34.2i)T + 2.40e3iT^{2} \)
11 \( 1 - 117T + 1.46e4T^{2} \)
13 \( 1 + (-51.4 + 51.4i)T - 2.85e4iT^{2} \)
17 \( 1 + (-180. - 180. i)T + 8.35e4iT^{2} \)
19 \( 1 + 595iT - 1.30e5T^{2} \)
23 \( 1 + (-124. + 124. i)T - 2.79e5iT^{2} \)
29 \( 1 - 1.17e3iT - 7.07e5T^{2} \)
31 \( 1 - 322T + 9.23e5T^{2} \)
37 \( 1 + (-578. - 578. i)T + 1.87e6iT^{2} \)
41 \( 1 + 63T + 2.82e6T^{2} \)
43 \( 1 + (1.25e3 - 1.25e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (462. + 462. i)T + 4.87e6iT^{2} \)
53 \( 1 + (1.49e3 - 1.49e3i)T - 7.89e6iT^{2} \)
59 \( 1 - 1.89e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.90e3T + 1.38e7T^{2} \)
67 \( 1 + (-4.77e3 - 4.77e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 2.68e3T + 2.54e7T^{2} \)
73 \( 1 + (-1.80e3 + 1.80e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 6.52e3iT - 3.89e7T^{2} \)
83 \( 1 + (-2.49e3 + 2.49e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 5.98e3iT - 6.27e7T^{2} \)
97 \( 1 + (3.46e3 + 3.46e3i)T + 8.85e7iT^{2} \)
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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

−16.80004802188542354731860994359, −14.81243402830806348020697384170, −13.55926761935484887877828546140, −12.45633006324168461182022136193, −10.90744702740400900573203950121, −9.484098105064359227894850138010, −8.384638158077992173462140959204, −6.83686431525319678094815536014, −3.08830934884094570981025646812, −1.17890803720598540804519342286, 3.55177669062008212425601689380, 6.22615711411197775727415396872, 8.063860975641847468927630909873, 9.242155114789224846266109708936, 9.829607761320255637309591271315, 12.15734380651498095213666869302, 14.16266496614102490181539025153, 15.19483078688064477215113443107, 16.06262624509800275303807014506, 16.93454102768579877087352344545

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