Properties

Label 2-7e2-49.25-c5-0-5
Degree $2$
Conductor $49$
Sign $-0.313 - 0.949i$
Analytic cond. $7.85880$
Root an. cond. $2.80335$
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  + (−7.64 + 2.35i)2-s + (2.34 + 5.97i)3-s + (26.5 − 18.0i)4-s + (−0.467 − 0.0704i)5-s + (−32.0 − 40.1i)6-s + (116. + 56.9i)7-s + (−0.443 + 0.556i)8-s + (147. − 137. i)9-s + (3.74 − 0.564i)10-s + (−107. − 99.8i)11-s + (170. + 115. i)12-s + (−183. + 803. i)13-s + (−1.02e3 − 160. i)14-s + (−0.674 − 2.95i)15-s + (−373. + 950. i)16-s + (−114. + 1.52e3i)17-s + ⋯
L(s)  = 1  + (−1.35 + 0.417i)2-s + (0.150 + 0.383i)3-s + (0.828 − 0.564i)4-s + (−0.00836 − 0.00126i)5-s + (−0.363 − 0.455i)6-s + (0.898 + 0.439i)7-s + (−0.00245 + 0.00307i)8-s + (0.608 − 0.565i)9-s + (0.0118 − 0.00178i)10-s + (−0.268 − 0.248i)11-s + (0.340 + 0.232i)12-s + (−0.300 + 1.31i)13-s + (−1.39 − 0.219i)14-s + (−0.000774 − 0.00339i)15-s + (−0.364 + 0.928i)16-s + (−0.0960 + 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.313 - 0.949i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -0.313 - 0.949i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.498430 + 0.689738i\)
\(L(\frac12)\) \(\approx\) \(0.498430 + 0.689738i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-116. - 56.9i)T \)
good2 \( 1 + (7.64 - 2.35i)T + (26.4 - 18.0i)T^{2} \)
3 \( 1 + (-2.34 - 5.97i)T + (-178. + 165. i)T^{2} \)
5 \( 1 + (0.467 + 0.0704i)T + (2.98e3 + 921. i)T^{2} \)
11 \( 1 + (107. + 99.8i)T + (1.20e4 + 1.60e5i)T^{2} \)
13 \( 1 + (183. - 803. i)T + (-3.34e5 - 1.61e5i)T^{2} \)
17 \( 1 + (114. - 1.52e3i)T + (-1.40e6 - 2.11e5i)T^{2} \)
19 \( 1 + (-206. - 358. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-2.84 - 37.9i)T + (-6.36e6 + 9.59e5i)T^{2} \)
29 \( 1 + (-4.63e3 + 2.23e3i)T + (1.27e7 - 1.60e7i)T^{2} \)
31 \( 1 + (4.72e3 - 8.17e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-9.59e3 - 6.54e3i)T + (2.53e7 + 6.45e7i)T^{2} \)
41 \( 1 + (-4.90e3 + 6.14e3i)T + (-2.57e7 - 1.12e8i)T^{2} \)
43 \( 1 + (-4.93e3 - 6.18e3i)T + (-3.27e7 + 1.43e8i)T^{2} \)
47 \( 1 + (7.18e3 - 2.21e3i)T + (1.89e8 - 1.29e8i)T^{2} \)
53 \( 1 + (1.90e4 - 1.29e4i)T + (1.52e8 - 3.89e8i)T^{2} \)
59 \( 1 + (1.20e4 - 1.82e3i)T + (6.83e8 - 2.10e8i)T^{2} \)
61 \( 1 + (1.35e4 + 9.24e3i)T + (3.08e8 + 7.86e8i)T^{2} \)
67 \( 1 + (-4.85e3 + 8.40e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-3.67e4 - 1.76e4i)T + (1.12e9 + 1.41e9i)T^{2} \)
73 \( 1 + (-4.07e4 - 1.25e4i)T + (1.71e9 + 1.16e9i)T^{2} \)
79 \( 1 + (-1.75e4 - 3.04e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (1.51e4 + 6.65e4i)T + (-3.54e9 + 1.70e9i)T^{2} \)
89 \( 1 + (4.32e4 - 4.01e4i)T + (4.17e8 - 5.56e9i)T^{2} \)
97 \( 1 - 8.63e4T + 8.58e9T^{2} \)
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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

−15.30341610591763595425990855837, −14.17486590279010435481155666374, −12.41804388410469132819535027928, −11.01920222624004513110737689539, −9.848886542738628265033817752665, −8.868738547528573708799946632112, −7.86510296962159571905239700450, −6.42690166667312100827393845231, −4.31546541913381955428374438815, −1.56739212377928710026141870936, 0.73032634678424383492331729676, 2.28003857775981493649920920090, 4.95514073421501744660608645176, 7.50680991534372372452480845632, 7.904615574013348672380831619199, 9.506103727373523920597108511690, 10.53383839894136781189421937077, 11.48547337313884341344090295652, 13.01217244639051823051726940650, 14.17436017188387442312794474797

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