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Properties

Label 2-5e4-25.4-c1-0-30
Degree $2$
Conductor $625$
Sign $-0.0627 + 0.998i$
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  + (1.07 + 0.350i)2-s + (1.52 − 2.10i)3-s + (−0.578 − 0.420i)4-s + (2.38 − 1.73i)6-s + 0.407i·7-s + (−1.80 − 2.48i)8-s + (−1.16 − 3.58i)9-s + (0.618 − 1.90i)11-s + (−1.76 + 0.574i)12-s + (−0.666 + 0.216i)13-s + (−0.142 + 0.439i)14-s + (−0.636 − 1.95i)16-s + (−0.930 − 1.28i)17-s − 4.27i·18-s + (4.00 − 2.90i)19-s + ⋯
L(s)  = 1  + (0.762 + 0.247i)2-s + (0.883 − 1.21i)3-s + (−0.289 − 0.210i)4-s + (0.974 − 0.708i)6-s + 0.153i·7-s + (−0.639 − 0.880i)8-s + (−0.388 − 1.19i)9-s + (0.186 − 0.573i)11-s + (−0.510 + 0.165i)12-s + (−0.184 + 0.0600i)13-s + (−0.0381 + 0.117i)14-s + (−0.159 − 0.489i)16-s + (−0.225 − 0.310i)17-s − 1.00i·18-s + (0.918 − 0.667i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.0627 + 0.998i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ -0.0627 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65881 - 1.76645i\)
\(L(\frac12)\) \(\approx\) \(1.65881 - 1.76645i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 - 0.350i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.52 + 2.10i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 - 0.407iT - 7T^{2} \)
11 \( 1 + (-0.618 + 1.90i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.666 - 0.216i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.930 + 1.28i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.00 + 2.90i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.14 + 0.371i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.45 - 3.23i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (6.63 - 4.82i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-4.88 + 1.58i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.22 - 6.86i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.16iT - 43T^{2} \)
47 \( 1 + (0.748 - 1.03i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.98 + 4.10i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.00 - 6.18i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.91 - 8.95i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-1.81 - 2.49i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (5.55 + 4.03i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.518 + 0.168i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.43 - 3.22i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.572 - 0.788i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.700 + 2.15i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-8.94 + 12.3i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.31078436573635364506633550287, −9.103477574981332038392908771075, −8.741777960741039121488149306586, −7.50350711035804061028649197273, −6.85441200152770630822845840776, −5.89619219181583489501829092214, −4.89624946941488157056685377671, −3.57368688620196898718683324028, −2.61023497879931567621194619115, −1.02431063576327737339719259524, 2.38151009724420803912576958671, 3.46593408972737583667434810120, 4.14201888295524261085303884440, 4.88991986170802038237310209626, 5.96171682968520138063746331650, 7.56644255604826111612852191965, 8.344577534669533833979365808555, 9.313760635176393341532030621133, 9.767988528600936675982676249389, 10.76556541037506700427164258254

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