Properties

Label 2-5e2-25.11-c5-0-1
Degree $2$
Conductor $25$
Sign $-0.870 + 0.492i$
Analytic cond. $4.00959$
Root an. cond. $2.00239$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.39 + 2.46i)2-s + (−4.95 + 15.2i)3-s + (−4.44 + 13.6i)4-s + (23.5 − 50.7i)5-s + (−20.7 − 63.9i)6-s − 192.·7-s + (−60.1 − 185. i)8-s + (−11.3 − 8.25i)9-s + (45.3 + 230. i)10-s + (−140. + 102. i)11-s + (−186. − 135. i)12-s + (−363. − 263. i)13-s + (654. − 475. i)14-s + (656. + 609. i)15-s + (288. + 209. i)16-s + (252. + 776. i)17-s + ⋯
L(s)  = 1  + (−0.600 + 0.436i)2-s + (−0.317 + 0.978i)3-s + (−0.138 + 0.427i)4-s + (0.420 − 0.907i)5-s + (−0.235 − 0.725i)6-s − 1.48·7-s + (−0.332 − 1.02i)8-s + (−0.0467 − 0.0339i)9-s + (0.143 + 0.727i)10-s + (−0.350 + 0.254i)11-s + (−0.374 − 0.271i)12-s + (−0.596 − 0.433i)13-s + (0.892 − 0.648i)14-s + (0.753 + 0.699i)15-s + (0.281 + 0.204i)16-s + (0.211 + 0.651i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.870 + 0.492i$
Analytic conductor: \(4.00959\)
Root analytic conductor: \(2.00239\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :5/2),\ -0.870 + 0.492i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0822359 - 0.312585i\)
\(L(\frac12)\) \(\approx\) \(0.0822359 - 0.312585i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-23.5 + 50.7i)T \)
good2 \( 1 + (3.39 - 2.46i)T + (9.88 - 30.4i)T^{2} \)
3 \( 1 + (4.95 - 15.2i)T + (-196. - 142. i)T^{2} \)
7 \( 1 + 192.T + 1.68e4T^{2} \)
11 \( 1 + (140. - 102. i)T + (4.97e4 - 1.53e5i)T^{2} \)
13 \( 1 + (363. + 263. i)T + (1.14e5 + 3.53e5i)T^{2} \)
17 \( 1 + (-252. - 776. i)T + (-1.14e6 + 8.34e5i)T^{2} \)
19 \( 1 + (-582. - 1.79e3i)T + (-2.00e6 + 1.45e6i)T^{2} \)
23 \( 1 + (3.52e3 - 2.55e3i)T + (1.98e6 - 6.12e6i)T^{2} \)
29 \( 1 + (795. - 2.44e3i)T + (-1.65e7 - 1.20e7i)T^{2} \)
31 \( 1 + (2.33e3 + 7.17e3i)T + (-2.31e7 + 1.68e7i)T^{2} \)
37 \( 1 + (-7.07e3 - 5.14e3i)T + (2.14e7 + 6.59e7i)T^{2} \)
41 \( 1 + (-3.45e3 - 2.51e3i)T + (3.58e7 + 1.10e8i)T^{2} \)
43 \( 1 - 2.78e3T + 1.47e8T^{2} \)
47 \( 1 + (2.99e3 - 9.21e3i)T + (-1.85e8 - 1.34e8i)T^{2} \)
53 \( 1 + (-6.65e3 + 2.04e4i)T + (-3.38e8 - 2.45e8i)T^{2} \)
59 \( 1 + (-2.62e4 - 1.90e4i)T + (2.20e8 + 6.79e8i)T^{2} \)
61 \( 1 + (1.14e4 - 8.35e3i)T + (2.60e8 - 8.03e8i)T^{2} \)
67 \( 1 + (1.55e4 + 4.77e4i)T + (-1.09e9 + 7.93e8i)T^{2} \)
71 \( 1 + (-1.28e4 + 3.96e4i)T + (-1.45e9 - 1.06e9i)T^{2} \)
73 \( 1 + (1.33e4 - 9.66e3i)T + (6.40e8 - 1.97e9i)T^{2} \)
79 \( 1 + (2.69e4 - 8.30e4i)T + (-2.48e9 - 1.80e9i)T^{2} \)
83 \( 1 + (-3.15e4 - 9.69e4i)T + (-3.18e9 + 2.31e9i)T^{2} \)
89 \( 1 + (4.18e4 - 3.04e4i)T + (1.72e9 - 5.31e9i)T^{2} \)
97 \( 1 + (5.21e3 - 1.60e4i)T + (-6.94e9 - 5.04e9i)T^{2} \)
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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

−16.75645356390476728314372316854, −16.40145548288051262424648750424, −15.43338154844507286018623533148, −13.21244783358594235007835669755, −12.37155327631932279345742377802, −9.898786645696678605871913179501, −9.632263534581112381178885722388, −7.85064218761165103887484936546, −5.81790417033938173000485212253, −3.87842821141767597786412418552, 0.26341560246492032878282220814, 2.49996392753780005959082244713, 6.02593342036038431655540803416, 7.11958669070916312936280158422, 9.387601683959826848439284103785, 10.32883787783638092178097071648, 11.81465528389084410752321832930, 13.20008176649585125333173212620, 14.31449183284262417605436796005, 15.96619272607258234829218884769

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