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Properties

Label 2-5e3-25.11-c1-0-0
Degree $2$
Conductor $125$
Sign $-0.990 - 0.137i$
Analytic cond. $0.998130$
Root an. cond. $0.999064$
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  + (−0.917 + 0.666i)2-s + (−0.804 + 2.47i)3-s + (−0.220 + 0.679i)4-s + (−0.911 − 2.80i)6-s + 0.407·7-s + (−0.951 − 2.92i)8-s + (−3.05 − 2.21i)9-s + (−1.61 + 1.17i)11-s + (−1.50 − 1.09i)12-s + (0.566 + 0.411i)13-s + (−0.373 + 0.271i)14-s + (1.66 + 1.21i)16-s + (0.489 + 1.50i)17-s + 4.27·18-s + (1.52 + 4.70i)19-s + ⋯
L(s)  = 1  + (−0.648 + 0.471i)2-s + (−0.464 + 1.42i)3-s + (−0.110 + 0.339i)4-s + (−0.372 − 1.14i)6-s + 0.153·7-s + (−0.336 − 1.03i)8-s + (−1.01 − 0.739i)9-s + (−0.487 + 0.354i)11-s + (−0.434 − 0.315i)12-s + (0.157 + 0.114i)13-s + (−0.0998 + 0.0725i)14-s + (0.416 + 0.302i)16-s + (0.118 + 0.365i)17-s + 1.00·18-s + (0.350 + 1.08i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $-0.990 - 0.137i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ -0.990 - 0.137i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0388534 + 0.561883i\)
\(L(\frac12)\) \(\approx\) \(0.0388534 + 0.561883i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.917 - 0.666i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.804 - 2.47i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 0.407T + 7T^{2} \)
11 \( 1 + (1.61 - 1.17i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.566 - 0.411i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.489 - 1.50i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.52 - 4.70i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.971 - 0.706i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.70 + 5.23i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.53 - 7.80i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.15 - 3.01i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.83 + 4.24i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.16T + 43T^{2} \)
47 \( 1 + (0.393 - 1.21i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.56 - 4.83i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.25 - 3.82i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.62 + 5.53i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.952 + 2.93i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.12 + 6.53i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.441 - 0.320i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.69 + 5.21i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.301 - 0.926i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-1.83 + 1.33i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.70 + 14.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.13348409212421909078163660076, −12.70152722952374420559104575012, −11.66324138772325817105690534601, −10.37043451061443990065211253544, −9.806849699338178080912938549310, −8.692278036030495307225291322749, −7.65190083315570062321452012239, −6.09966443940701535203330854605, −4.71412342736530764822719580512, −3.54886097994024641580863415663, 0.810593501410645967593817049504, 2.43203586664324826657390650348, 5.19619436881263389200785992736, 6.31395225664131490545313988156, 7.56497996808848274240851426322, 8.567259005747043526515145635603, 9.820456649446555237233300585985, 11.10137195684662197579160489602, 11.63864961503941698209005038477, 12.87011966128533994981866109173

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