Properties

Label 2-5e2-25.14-c5-0-3
Degree $2$
Conductor $25$
Sign $-0.187 - 0.982i$
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  + (4.57 + 6.30i)2-s + (3.07 + 0.997i)3-s + (−8.86 + 27.2i)4-s + (10.4 + 54.9i)5-s + (7.77 + 23.9i)6-s + 46.1i·7-s + (24.4 − 7.93i)8-s + (−188. − 136. i)9-s + (−298. + 317. i)10-s + (144. − 104. i)11-s + (−54.4 + 74.9i)12-s + (267. − 368. i)13-s + (−290. + 211. i)14-s + (−22.5 + 179. i)15-s + (904. + 657. i)16-s + (1.37e3 − 445. i)17-s + ⋯
L(s)  = 1  + (0.809 + 1.11i)2-s + (0.196 + 0.0640i)3-s + (−0.277 + 0.853i)4-s + (0.187 + 0.982i)5-s + (0.0881 + 0.271i)6-s + 0.355i·7-s + (0.134 − 0.0438i)8-s + (−0.774 − 0.562i)9-s + (−0.942 + 1.00i)10-s + (0.358 − 0.260i)11-s + (−0.109 + 0.150i)12-s + (0.439 − 0.605i)13-s + (−0.396 + 0.288i)14-s + (−0.0259 + 0.205i)15-s + (0.883 + 0.642i)16-s + (1.15 − 0.374i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.187 - 0.982i$
Analytic conductor: \(4.00959\)
Root analytic conductor: \(2.00239\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :5/2),\ -0.187 - 0.982i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.47523 + 1.78367i\)
\(L(\frac12)\) \(\approx\) \(1.47523 + 1.78367i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-10.4 - 54.9i)T \)
good2 \( 1 + (-4.57 - 6.30i)T + (-9.88 + 30.4i)T^{2} \)
3 \( 1 + (-3.07 - 0.997i)T + (196. + 142. i)T^{2} \)
7 \( 1 - 46.1iT - 1.68e4T^{2} \)
11 \( 1 + (-144. + 104. i)T + (4.97e4 - 1.53e5i)T^{2} \)
13 \( 1 + (-267. + 368. i)T + (-1.14e5 - 3.53e5i)T^{2} \)
17 \( 1 + (-1.37e3 + 445. i)T + (1.14e6 - 8.34e5i)T^{2} \)
19 \( 1 + (68.7 + 211. i)T + (-2.00e6 + 1.45e6i)T^{2} \)
23 \( 1 + (2.09e3 + 2.88e3i)T + (-1.98e6 + 6.12e6i)T^{2} \)
29 \( 1 + (269. - 830. i)T + (-1.65e7 - 1.20e7i)T^{2} \)
31 \( 1 + (-520. - 1.60e3i)T + (-2.31e7 + 1.68e7i)T^{2} \)
37 \( 1 + (8.11e3 - 1.11e4i)T + (-2.14e7 - 6.59e7i)T^{2} \)
41 \( 1 + (8.74e3 + 6.35e3i)T + (3.58e7 + 1.10e8i)T^{2} \)
43 \( 1 + 2.74e3iT - 1.47e8T^{2} \)
47 \( 1 + (2.11e4 + 6.88e3i)T + (1.85e8 + 1.34e8i)T^{2} \)
53 \( 1 + (6.80e3 + 2.21e3i)T + (3.38e8 + 2.45e8i)T^{2} \)
59 \( 1 + (-3.72e4 - 2.70e4i)T + (2.20e8 + 6.79e8i)T^{2} \)
61 \( 1 + (1.60e4 - 1.16e4i)T + (2.60e8 - 8.03e8i)T^{2} \)
67 \( 1 + (-4.75e4 + 1.54e4i)T + (1.09e9 - 7.93e8i)T^{2} \)
71 \( 1 + (1.32e4 - 4.09e4i)T + (-1.45e9 - 1.06e9i)T^{2} \)
73 \( 1 + (2.83e4 + 3.90e4i)T + (-6.40e8 + 1.97e9i)T^{2} \)
79 \( 1 + (-2.72e4 + 8.37e4i)T + (-2.48e9 - 1.80e9i)T^{2} \)
83 \( 1 + (-6.30e4 + 2.04e4i)T + (3.18e9 - 2.31e9i)T^{2} \)
89 \( 1 + (8.13e4 - 5.90e4i)T + (1.72e9 - 5.31e9i)T^{2} \)
97 \( 1 + (1.98e4 + 6.43e3i)T + (6.94e9 + 5.04e9i)T^{2} \)
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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

−16.57462333511242551033736029418, −15.29163003800170788137334956490, −14.52609561155957846571995601408, −13.70795688193726084861820993601, −11.98103002041820090721824193480, −10.28119870054273635177895742015, −8.315205002008278030124962512110, −6.69527638319911027584210234639, −5.62405521237971451171633024990, −3.36227108266422030002658160991, 1.68160506342705082412867025970, 3.82143841961905378969539658673, 5.41591001495219990642070864112, 8.069121877071693991149859759028, 9.769864750020051334132385777690, 11.31858402086396710044249847728, 12.34415992798999906549653080713, 13.51910683103467312133700099974, 14.29140593580586247876907759772, 16.33377015443421664870048098063

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