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Properties

Label 2-5e4-25.11-c1-0-18
Degree $2$
Conductor $625$
Sign $0.855 - 0.518i$
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  + (−1.62 + 1.18i)2-s + (−0.934 + 2.87i)3-s + (0.634 − 1.95i)4-s + (−1.88 − 5.79i)6-s − 0.369·7-s + (0.0341 + 0.105i)8-s + (−4.97 − 3.61i)9-s + (1.41 − 1.02i)11-s + (5.02 + 3.65i)12-s + (0.903 + 0.656i)13-s + (0.602 − 0.437i)14-s + (3.14 + 2.28i)16-s + (−1.69 − 5.21i)17-s + 12.3·18-s + (−1.15 − 3.56i)19-s + ⋯
L(s)  = 1  + (−1.15 + 0.836i)2-s + (−0.539 + 1.66i)3-s + (0.317 − 0.977i)4-s + (−0.768 − 2.36i)6-s − 0.139·7-s + (0.0120 + 0.0371i)8-s + (−1.65 − 1.20i)9-s + (0.425 − 0.309i)11-s + (1.45 + 1.05i)12-s + (0.250 + 0.181i)13-s + (0.161 − 0.117i)14-s + (0.786 + 0.571i)16-s + (−0.411 − 1.26i)17-s + 2.92·18-s + (−0.266 − 0.818i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.855 - 0.518i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.855 - 0.518i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.390061 + 0.108907i\)
\(L(\frac12)\) \(\approx\) \(0.390061 + 0.108907i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 1.18i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.934 - 2.87i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 0.369T + 7T^{2} \)
11 \( 1 + (-1.41 + 1.02i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.903 - 0.656i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.69 + 5.21i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.15 + 3.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-5.86 + 4.25i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.29 + 3.98i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.0944 - 0.290i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.45 + 5.41i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.38 - 2.46i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.17T + 43T^{2} \)
47 \( 1 + (0.250 - 0.770i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.20 + 3.72i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.50 - 1.09i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (7.83 - 5.69i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.85 - 11.8i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.60 + 11.1i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.77 + 2.01i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.75 + 5.41i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.20 + 6.78i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (1.23 - 0.899i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (1.97 - 6.08i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.47600340787935451250472230635, −9.539013762868162838155119642816, −9.109787547979672357358129897166, −8.446614255013571271009637735445, −7.03956597363425187421419255619, −6.34583744664466925755690093795, −5.23766624300852537592438146991, −4.34460326359987247825245899310, −3.13192660622961875801247426667, −0.39055583812199351510146633567, 1.25434578451315930045501411256, 1.90580046208649946605001678315, 3.32268802528149645136299469650, 5.29977528240406087476846446930, 6.34180754592084332920363354637, 7.10957643302321283255373976088, 8.110649149693185350791653678298, 8.621523688646433326747928963190, 9.722090660511339408333945681823, 10.76337716828239880900536049072

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