Properties

Label 2-7e2-49.25-c5-0-7
Degree $2$
Conductor $49$
Sign $0.458 - 0.888i$
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  + (−6.81 + 2.10i)2-s + (3.52 + 8.98i)3-s + (15.5 − 10.6i)4-s + (80.2 + 12.0i)5-s + (−42.9 − 53.8i)6-s + (−85.3 − 97.5i)7-s + (58.3 − 73.2i)8-s + (109. − 101. i)9-s + (−572. + 86.2i)10-s + (415. + 385. i)11-s + (150. + 102. i)12-s + (20.6 − 90.5i)13-s + (786. + 485. i)14-s + (174. + 763. i)15-s + (−464. + 1.18e3i)16-s + (−5.71 + 76.3i)17-s + ⋯
L(s)  = 1  + (−1.20 + 0.371i)2-s + (0.226 + 0.576i)3-s + (0.487 − 0.332i)4-s + (1.43 + 0.216i)5-s + (−0.486 − 0.610i)6-s + (−0.658 − 0.752i)7-s + (0.322 − 0.404i)8-s + (0.452 − 0.419i)9-s + (−1.81 + 0.272i)10-s + (1.03 + 0.961i)11-s + (0.301 + 0.205i)12-s + (0.0339 − 0.148i)13-s + (1.07 + 0.662i)14-s + (0.199 + 0.876i)15-s + (−0.453 + 1.15i)16-s + (−0.00480 + 0.0640i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.458 - 0.888i$
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: \((2,\ 49,\ (\ :5/2),\ 0.458 - 0.888i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.03175 + 0.629006i\)
\(L(\frac12)\) \(\approx\) \(1.03175 + 0.629006i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (85.3 + 97.5i)T \)
good2 \( 1 + (6.81 - 2.10i)T + (26.4 - 18.0i)T^{2} \)
3 \( 1 + (-3.52 - 8.98i)T + (-178. + 165. i)T^{2} \)
5 \( 1 + (-80.2 - 12.0i)T + (2.98e3 + 921. i)T^{2} \)
11 \( 1 + (-415. - 385. i)T + (1.20e4 + 1.60e5i)T^{2} \)
13 \( 1 + (-20.6 + 90.5i)T + (-3.34e5 - 1.61e5i)T^{2} \)
17 \( 1 + (5.71 - 76.3i)T + (-1.40e6 - 2.11e5i)T^{2} \)
19 \( 1 + (-1.25e3 - 2.17e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (51.6 + 688. i)T + (-6.36e6 + 9.59e5i)T^{2} \)
29 \( 1 + (5.08e3 - 2.44e3i)T + (1.27e7 - 1.60e7i)T^{2} \)
31 \( 1 + (-3.35e3 + 5.81e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (5.42e3 + 3.70e3i)T + (2.53e7 + 6.45e7i)T^{2} \)
41 \( 1 + (6.18e3 - 7.75e3i)T + (-2.57e7 - 1.12e8i)T^{2} \)
43 \( 1 + (-1.26e4 - 1.59e4i)T + (-3.27e7 + 1.43e8i)T^{2} \)
47 \( 1 + (-2.64e4 + 8.14e3i)T + (1.89e8 - 1.29e8i)T^{2} \)
53 \( 1 + (-9.74e3 + 6.64e3i)T + (1.52e8 - 3.89e8i)T^{2} \)
59 \( 1 + (-772. + 116. i)T + (6.83e8 - 2.10e8i)T^{2} \)
61 \( 1 + (8.39e3 + 5.72e3i)T + (3.08e8 + 7.86e8i)T^{2} \)
67 \( 1 + (-1.77e4 + 3.07e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (7.59e4 + 3.65e4i)T + (1.12e9 + 1.41e9i)T^{2} \)
73 \( 1 + (-2.96e3 - 915. i)T + (1.71e9 + 1.16e9i)T^{2} \)
79 \( 1 + (-3.55e4 - 6.15e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (6.67e3 + 2.92e4i)T + (-3.54e9 + 1.70e9i)T^{2} \)
89 \( 1 + (-2.50e4 + 2.32e4i)T + (4.17e8 - 5.56e9i)T^{2} \)
97 \( 1 + 1.49e4T + 8.58e9T^{2} \)
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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

−14.95012287563337976863800466727, −13.82513462335979505827002686920, −12.61928210611627691211107483991, −10.40000102931515129069053644997, −9.742328560162704864703724388427, −9.259523545015295441130437148406, −7.35219772705841496839439015255, −6.26523764149330729021163442185, −3.95390124495980555456368792798, −1.39195651121882125395974784293, 1.13827723774210951766815213662, 2.42254152244723579550291360809, 5.55118257458697341902742877620, 7.00531087601936105775652735769, 8.827589950428086613395987593493, 9.299918447144181434523750764214, 10.47730645994847030592896806255, 11.92585590564180699684439246896, 13.45251216233722796060576400365, 13.92274202984052550002412191414

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