Properties

Label 2-5e2-25.22-c2-0-0
Degree $2$
Conductor $25$
Sign $-0.931 - 0.364i$
Analytic cond. $0.681200$
Root an. cond. $0.825348$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.38 + 1.21i)2-s + (−3.57 + 0.566i)3-s + (1.85 − 2.54i)4-s + (−4.45 + 2.27i)5-s + (7.83 − 5.69i)6-s + (6.54 + 6.54i)7-s + (0.355 − 2.24i)8-s + (3.91 − 1.27i)9-s + (7.83 − 10.8i)10-s + (−3.18 + 9.80i)11-s + (−5.17 + 10.1i)12-s + (−11.3 − 5.77i)13-s + (−23.5 − 7.65i)14-s + (14.6 − 10.6i)15-s + (5.77 + 17.7i)16-s + (0.578 + 0.0915i)17-s + ⋯
L(s)  = 1  + (−1.19 + 0.606i)2-s + (−1.19 + 0.188i)3-s + (0.462 − 0.636i)4-s + (−0.890 + 0.455i)5-s + (1.30 − 0.948i)6-s + (0.935 + 0.935i)7-s + (0.0444 − 0.280i)8-s + (0.434 − 0.141i)9-s + (0.783 − 1.08i)10-s + (−0.289 + 0.891i)11-s + (−0.431 + 0.846i)12-s + (−0.871 − 0.444i)13-s + (−1.68 − 0.546i)14-s + (0.975 − 0.711i)15-s + (0.360 + 1.11i)16-s + (0.0340 + 0.00538i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.931 - 0.364i$
Analytic conductor: \(0.681200\)
Root analytic conductor: \(0.825348\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :1),\ -0.931 - 0.364i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0471229 + 0.249871i\)
\(L(\frac12)\) \(\approx\) \(0.0471229 + 0.249871i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.45 - 2.27i)T \)
good2 \( 1 + (2.38 - 1.21i)T + (2.35 - 3.23i)T^{2} \)
3 \( 1 + (3.57 - 0.566i)T + (8.55 - 2.78i)T^{2} \)
7 \( 1 + (-6.54 - 6.54i)T + 49iT^{2} \)
11 \( 1 + (3.18 - 9.80i)T + (-97.8 - 71.1i)T^{2} \)
13 \( 1 + (11.3 + 5.77i)T + (99.3 + 136. i)T^{2} \)
17 \( 1 + (-0.578 - 0.0915i)T + (274. + 89.3i)T^{2} \)
19 \( 1 + (-1.57 - 2.16i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + (-16.5 - 32.4i)T + (-310. + 427. i)T^{2} \)
29 \( 1 + (-9.15 + 12.6i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (15.1 - 10.9i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (26.1 - 51.3i)T + (-804. - 1.10e3i)T^{2} \)
41 \( 1 + (-0.808 - 2.48i)T + (-1.35e3 + 988. i)T^{2} \)
43 \( 1 + (7.76 - 7.76i)T - 1.84e3iT^{2} \)
47 \( 1 + (6.13 + 38.7i)T + (-2.10e3 + 682. i)T^{2} \)
53 \( 1 + (-39.0 + 6.18i)T + (2.67e3 - 868. i)T^{2} \)
59 \( 1 + (-0.494 + 0.160i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-7.44 + 22.9i)T + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + (-6.71 - 1.06i)T + (4.26e3 + 1.38e3i)T^{2} \)
71 \( 1 + (-80.2 - 58.3i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-27.3 - 53.6i)T + (-3.13e3 + 4.31e3i)T^{2} \)
79 \( 1 + (-2.28 + 3.14i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (10.3 - 65.4i)T + (-6.55e3 - 2.12e3i)T^{2} \)
89 \( 1 + (49.5 + 16.0i)T + (6.40e3 + 4.65e3i)T^{2} \)
97 \( 1 + (-0.610 - 3.85i)T + (-8.94e3 + 2.90e3i)T^{2} \)
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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

−17.76881755723244691635609175996, −17.02701138745524775739740952990, −15.64447360631118295264181499855, −15.02082992755766604770966463697, −12.29607418552064484739334334592, −11.32490820185808803951179099993, −10.01690132627934471679503044316, −8.267769542691984107898920695577, −7.07494462927535818776735728409, −5.15639626532002690639409012138, 0.59588774193697738264585905627, 4.93602564596353106221573162120, 7.39178216663454693823579173031, 8.707564344676892490921017461846, 10.68625275369581608284251028995, 11.23785056132812567603126528119, 12.32760316387274247302662145855, 14.37389738562518082590537038238, 16.47090080243887852246291482111, 16.97955442545312657594765490447

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