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Properties

Label 2-5e4-125.104-c1-0-1
Degree $2$
Conductor $625$
Sign $-0.946 - 0.323i$
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  + (−0.0406 − 0.158i)2-s + (−1.60 + 0.306i)3-s + (1.72 − 0.950i)4-s + (0.113 + 0.241i)6-s + (−0.875 + 1.20i)7-s + (−0.444 − 0.473i)8-s + (−0.305 + 0.121i)9-s + (−4.14 + 1.06i)11-s + (−2.48 + 2.05i)12-s + (−1.43 − 3.61i)13-s + (0.226 + 0.0895i)14-s + (2.05 − 3.24i)16-s + (−1.44 + 2.62i)17-s + (0.0315 + 0.0434i)18-s + (−0.997 + 5.23i)19-s + ⋯
L(s)  = 1  + (−0.0287 − 0.111i)2-s + (−0.926 + 0.176i)3-s + (0.864 − 0.475i)4-s + (0.0464 + 0.0986i)6-s + (−0.330 + 0.455i)7-s + (−0.157 − 0.167i)8-s + (−0.101 + 0.0403i)9-s + (−1.25 + 0.321i)11-s + (−0.717 + 0.593i)12-s + (−0.397 − 1.00i)13-s + (0.0604 + 0.0239i)14-s + (0.514 − 0.810i)16-s + (−0.350 + 0.637i)17-s + (0.00744 + 0.0102i)18-s + (−0.228 + 1.20i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.946 - 0.323i$
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: \((2,\ 625,\ (\ :1/2),\ -0.946 - 0.323i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0163267 + 0.0982876i\)
\(L(\frac12)\) \(\approx\) \(0.0163267 + 0.0982876i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0406 + 0.158i)T + (-1.75 + 0.963i)T^{2} \)
3 \( 1 + (1.60 - 0.306i)T + (2.78 - 1.10i)T^{2} \)
7 \( 1 + (0.875 - 1.20i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (4.14 - 1.06i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (1.43 + 3.61i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (1.44 - 2.62i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (0.997 - 5.23i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (2.26 + 0.142i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (4.03 + 0.509i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (7.84 + 4.31i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (-5.64 - 3.58i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (0.106 + 1.69i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (3.75 - 1.22i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (8.41 - 8.95i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (8.78 + 4.13i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (-2.23 - 2.70i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (-0.0370 + 0.589i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-0.458 - 3.62i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (-0.992 - 0.932i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (3.33 + 2.76i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (2.80 + 14.6i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (2.85 + 0.543i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-0.276 + 0.333i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (-1.52 + 12.0i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.94062069916187947185313449005, −10.30493466727443749828371787456, −9.689220301156702685530539532582, −8.167872625035702419722518150226, −7.47359074028313657669582351862, −5.99281658454628516664025670427, −5.89626526126915398000751249735, −4.83684409235128825621855844001, −3.13806534481077373525059118351, −2.01869192256307049402270078437, 0.05409605029103063256257197408, 2.20128715764010357382393265316, 3.35905874515791412957929725844, 4.83705997598499620546343667320, 5.76512991498738655235192752249, 6.81624906783347371354998670024, 7.18400035806222407549649690641, 8.331195326362531298939464008323, 9.389213917941778660187607217735, 10.55938914790229011811537263014

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