Properties

Label 2-5e4-25.9-c1-0-23
Degree $2$
Conductor $625$
Sign $-0.425 + 0.904i$
Analytic cond. $4.99065$
Root an. cond. $2.23397$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.107 + 0.148i)2-s + (−1.39 + 0.454i)3-s + (0.607 + 1.87i)4-s + (0.0833 − 0.256i)6-s − 3.26i·7-s + (−0.690 − 0.224i)8-s + (−0.674 + 0.489i)9-s + (−1.61 − 1.17i)11-s + (−1.70 − 2.34i)12-s + (−0.174 − 0.239i)13-s + (0.483 + 0.351i)14-s + (−3.07 + 2.23i)16-s + (−4.91 − 1.59i)17-s − 0.152i·18-s + (−0.534 + 1.64i)19-s + ⋯
L(s)  = 1  + (−0.0761 + 0.104i)2-s + (−0.808 + 0.262i)3-s + (0.303 + 0.935i)4-s + (0.0340 − 0.104i)6-s − 1.23i·7-s + (−0.244 − 0.0793i)8-s + (−0.224 + 0.163i)9-s + (−0.487 − 0.354i)11-s + (−0.491 − 0.675i)12-s + (−0.0483 − 0.0665i)13-s + (0.129 + 0.0938i)14-s + (−0.768 + 0.558i)16-s + (−1.19 − 0.386i)17-s − 0.0359i·18-s + (−0.122 + 0.377i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.425 + 0.904i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ -0.425 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.143336 - 0.225861i\)
\(L(\frac12)\) \(\approx\) \(0.143336 - 0.225861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (0.107 - 0.148i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (1.39 - 0.454i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + 3.26iT - 7T^{2} \)
11 \( 1 + (1.61 + 1.17i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.174 + 0.239i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.91 + 1.59i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.534 - 1.64i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.516 - 0.711i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.82 + 5.62i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.88 + 5.80i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.75 + 6.54i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.821 + 0.596i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.24iT - 43T^{2} \)
47 \( 1 + (4.01 - 1.30i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.70 - 2.50i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.80 + 3.48i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.740 + 0.538i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-6.55 - 2.12i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.84 + 5.67i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.19 - 7.14i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.39 - 7.38i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (13.8 + 4.48i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-6.08 - 4.42i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-6.39 + 2.07i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.61459597030078652136298809301, −9.567220334301982380755104538064, −8.352952590623481374160610766693, −7.67482149901314812835848886855, −6.78715235677889479473332422829, −5.88082048081547192359891099093, −4.62465838329321984475709419802, −3.81590858635048340934948757747, −2.45577024393013796194616213774, −0.15459992459829442774747433099, 1.71873469978833849989652064776, 2.86196695945995320566698408567, 4.85595445568574415413349434392, 5.45090542808218883917386457405, 6.36665756855456053406474977365, 6.92295944783999293192582054429, 8.544296667620770209078006714452, 9.108861288039614094039913382108, 10.21627174010747110968262087475, 10.95435362262085235434396006899

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