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Properties

Label 2-5e2-5.3-c4-0-2
Degree $2$
Conductor $25$
Sign $0.525 + 0.850i$
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  + (−4.58 + 4.58i)2-s + (−4.58 − 4.58i)3-s − 26i·4-s + 42·6-s + (41.2 − 41.2i)7-s + (45.8 + 45.8i)8-s − 39i·9-s − 108·11-s + (−119. + 119. i)12-s + (−164. − 164. i)13-s + 378i·14-s − 3.99·16-s + (18.3 − 18.3i)17-s + (178. + 178. i)18-s − 140i·19-s + ⋯
L(s)  = 1  + (−1.14 + 1.14i)2-s + (−0.509 − 0.509i)3-s − 1.62i·4-s + 1.16·6-s + (0.841 − 0.841i)7-s + (0.716 + 0.716i)8-s − 0.481i·9-s − 0.892·11-s + (−0.827 + 0.827i)12-s + (−0.976 − 0.976i)13-s + 1.92i·14-s − 0.0156·16-s + (0.0634 − 0.0634i)17-s + (0.551 + 0.551i)18-s − 0.387i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(2.58424\)
Root analytic conductor: \(1.60755\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :2),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.412006 - 0.229708i\)
\(L(\frac12)\) \(\approx\) \(0.412006 - 0.229708i\)
\(L(3)\) 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 \( 1 + (4.58 - 4.58i)T - 16iT^{2} \)
3 \( 1 + (4.58 + 4.58i)T + 81iT^{2} \)
7 \( 1 + (-41.2 + 41.2i)T - 2.40e3iT^{2} \)
11 \( 1 + 108T + 1.46e4T^{2} \)
13 \( 1 + (164. + 164. i)T + 2.85e4iT^{2} \)
17 \( 1 + (-18.3 + 18.3i)T - 8.35e4iT^{2} \)
19 \( 1 + 140iT - 1.30e5T^{2} \)
23 \( 1 + (-362. - 362. i)T + 2.79e5iT^{2} \)
29 \( 1 + 810iT - 7.07e5T^{2} \)
31 \( 1 + 728T + 9.23e5T^{2} \)
37 \( 1 + (-659. + 659. i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.51e3T + 2.82e6T^{2} \)
43 \( 1 + (-41.2 - 41.2i)T + 3.41e6iT^{2} \)
47 \( 1 + (-178. + 178. i)T - 4.87e6iT^{2} \)
53 \( 1 + (3.09e3 + 3.09e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 3.78e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.59e3T + 1.38e7T^{2} \)
67 \( 1 + (-3.34e3 + 3.34e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 432T + 2.54e7T^{2} \)
73 \( 1 + (-6.43e3 - 6.43e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 8.84e3iT - 3.89e7T^{2} \)
83 \( 1 + (-774. - 774. i)T + 4.74e7iT^{2} \)
89 \( 1 + 1.32e4iT - 6.27e7T^{2} \)
97 \( 1 + (-8.08e3 + 8.08e3i)T - 8.85e7iT^{2} \)
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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

−17.09894093902133459992741657963, −15.60816698518083049779354752117, −14.59598018809083243655301646492, −12.85252765130347264372541025863, −11.05307183057885953789274121068, −9.696939734558382659396119093107, −7.934702921357344420241828112325, −7.14420310823011059125067556104, −5.45450962887272460739985452436, −0.58416573543864551858608430798, 2.23756114453069697122663772602, 5.04571466633359189653701663886, 7.921994448892469209051337068858, 9.264688062490167833544554826785, 10.56098813111916343885663426486, 11.41303455750850584607831408048, 12.53624717842331331620529175073, 14.64436431599160315721799497311, 16.27402236492210280993147414944, 17.28700139792589992921503953429

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