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Properties

Label 2-5e4-125.104-c1-0-23
Degree $2$
Conductor $625$
Sign $0.991 + 0.129i$
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  + (0.0930 + 0.362i)2-s + (1.71 − 0.327i)3-s + (1.62 − 0.896i)4-s + (0.278 + 0.592i)6-s + (−0.0200 + 0.0276i)7-s + (0.989 + 1.05i)8-s + (0.0572 − 0.0226i)9-s + (3.09 − 0.795i)11-s + (2.50 − 2.07i)12-s + (−0.523 − 1.32i)13-s + (−0.0118 − 0.00470i)14-s + (1.70 − 2.68i)16-s + (−2.23 + 4.06i)17-s + (0.0135 + 0.0186i)18-s + (0.155 − 0.814i)19-s + ⋯
L(s)  = 1  + (0.0658 + 0.256i)2-s + (0.992 − 0.189i)3-s + (0.814 − 0.448i)4-s + (0.113 + 0.241i)6-s + (−0.00758 + 0.0104i)7-s + (0.349 + 0.372i)8-s + (0.0190 − 0.00755i)9-s + (0.934 − 0.239i)11-s + (0.723 − 0.598i)12-s + (−0.145 − 0.366i)13-s + (−0.00317 − 0.00125i)14-s + (0.425 − 0.671i)16-s + (−0.542 + 0.986i)17-s + (0.00319 + 0.00439i)18-s + (0.0356 − 0.186i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.991 + 0.129i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.991 + 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.54706 - 0.165494i\)
\(L(\frac12)\) \(\approx\) \(2.54706 - 0.165494i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0930 - 0.362i)T + (-1.75 + 0.963i)T^{2} \)
3 \( 1 + (-1.71 + 0.327i)T + (2.78 - 1.10i)T^{2} \)
7 \( 1 + (0.0200 - 0.0276i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (-3.09 + 0.795i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (0.523 + 1.32i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (2.23 - 4.06i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (-0.155 + 0.814i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-3.59 - 0.226i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (-4.73 - 0.598i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (5.09 + 2.80i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (2.33 + 1.48i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (0.697 + 11.0i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (8.03 - 2.60i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (6.03 - 6.43i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (-1.36 - 0.644i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (7.34 + 8.88i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (-0.0901 + 1.43i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-1.31 - 10.3i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (-4.44 - 4.17i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (2.19 + 1.81i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (-2.45 - 12.8i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (-9.53 - 1.81i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (2.41 - 2.92i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (-1.59 + 12.6i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.70106739970272393858853333333, −9.567489528775502844337602858991, −8.747615117906844483873962333656, −7.969432888072068525989380904686, −7.03218333927115428543497612310, −6.25318060100888904758630331486, −5.21436448280589740780191301873, −3.75140934170344558511899119075, −2.67439392937707486611737470851, −1.57470237428922812878174956784, 1.77015633189278250376348177273, 2.89990113920054674483924007082, 3.65349613651829650167275841816, 4.83490778154159984449375443822, 6.45620642992398730786166906365, 7.06055191592330826408310398942, 8.091961660636062022636162379836, 8.915978039981708378140904980240, 9.622881543916146795057729619765, 10.64510125619515777515711239384

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