Properties

Label 2-7e2-49.2-c5-0-16
Degree $2$
Conductor $49$
Sign $0.991 - 0.129i$
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  + (9.25 + 2.85i)2-s + (4.05 − 10.3i)3-s + (51.0 + 34.8i)4-s + (20.5 − 3.09i)5-s + (67.0 − 84.0i)6-s + (13.7 − 128. i)7-s + (180. + 225. i)8-s + (87.8 + 81.5i)9-s + (199. + 30.0i)10-s + (−385. + 357. i)11-s + (566. − 386. i)12-s + (67.0 + 293. i)13-s + (494. − 1.15e3i)14-s + (51.3 − 224. i)15-s + (299. + 763. i)16-s + (−109. − 1.45e3i)17-s + ⋯
L(s)  = 1  + (1.63 + 0.504i)2-s + (0.260 − 0.662i)3-s + (1.59 + 1.08i)4-s + (0.367 − 0.0554i)5-s + (0.759 − 0.952i)6-s + (0.105 − 0.994i)7-s + (0.995 + 1.24i)8-s + (0.361 + 0.335i)9-s + (0.629 + 0.0948i)10-s + (−0.959 + 0.890i)11-s + (1.13 − 0.774i)12-s + (0.110 + 0.482i)13-s + (0.674 − 1.57i)14-s + (0.0588 − 0.257i)15-s + (0.292 + 0.745i)16-s + (−0.0915 − 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.991 - 0.129i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ 0.991 - 0.129i)\)

Particular Values

\(L(3)\) \(\approx\) \(4.35398 + 0.283726i\)
\(L(\frac12)\) \(\approx\) \(4.35398 + 0.283726i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-13.7 + 128. i)T \)
good2 \( 1 + (-9.25 - 2.85i)T + (26.4 + 18.0i)T^{2} \)
3 \( 1 + (-4.05 + 10.3i)T + (-178. - 165. i)T^{2} \)
5 \( 1 + (-20.5 + 3.09i)T + (2.98e3 - 921. i)T^{2} \)
11 \( 1 + (385. - 357. i)T + (1.20e4 - 1.60e5i)T^{2} \)
13 \( 1 + (-67.0 - 293. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (109. + 1.45e3i)T + (-1.40e6 + 2.11e5i)T^{2} \)
19 \( 1 + (633. - 1.09e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (158. - 2.11e3i)T + (-6.36e6 - 9.59e5i)T^{2} \)
29 \( 1 + (5.33e3 + 2.56e3i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (-510. - 884. i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (1.39e3 - 949. i)T + (2.53e7 - 6.45e7i)T^{2} \)
41 \( 1 + (2.68e3 + 3.36e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (-4.84e3 + 6.07e3i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (-1.00e3 - 308. i)T + (1.89e8 + 1.29e8i)T^{2} \)
53 \( 1 + (-1.99e4 - 1.36e4i)T + (1.52e8 + 3.89e8i)T^{2} \)
59 \( 1 + (-4.70e4 - 7.09e3i)T + (6.83e8 + 2.10e8i)T^{2} \)
61 \( 1 + (-2.58e4 + 1.76e4i)T + (3.08e8 - 7.86e8i)T^{2} \)
67 \( 1 + (-2.85e4 - 4.95e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-5.54e3 + 2.66e3i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (6.11e4 - 1.88e4i)T + (1.71e9 - 1.16e9i)T^{2} \)
79 \( 1 + (-2.21e4 + 3.83e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-5.23e3 + 2.29e4i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (-5.80e4 - 5.38e4i)T + (4.17e8 + 5.56e9i)T^{2} \)
97 \( 1 + 1.36e5T + 8.58e9T^{2} \)
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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

−14.22365526085790030493829023322, −13.48147561734409035154253453183, −12.93373976166115947634275152596, −11.63319611921205731200393807236, −10.03305065670993526995903042684, −7.61773133271261857662329797947, −7.02983997322858557355775593379, −5.37535379291637611765258416882, −4.08671953974348019505393736069, −2.14114242386568898436191927910, 2.38384116699271704973125402440, 3.71346302110558326686306324456, 5.20362711371285068179039740274, 6.20128932609787519886083373259, 8.595260484265271386893881524086, 10.23666793401303449373532248281, 11.22078590018141783896329708033, 12.61384586218007398630566709889, 13.25441523979897015680998986856, 14.65188656175597168387919031153

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