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Properties

Label 2-7e2-7.4-c5-0-14
Degree $2$
Conductor $49$
Sign $-0.605 - 0.795i$
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  + (−4.13 − 7.16i)2-s + (12.8 − 22.2i)3-s + (−18.2 + 31.5i)4-s + (−14.3 − 24.8i)5-s − 212.·6-s + 37.0·8-s + (−207. − 359. i)9-s + (−118. + 206. i)10-s + (135. − 233. i)11-s + (467. + 810. i)12-s + 300.·13-s − 737.·15-s + (430. + 745. i)16-s + (−306. + 530. i)17-s + (−1.71e3 + 2.97e3i)18-s + (850. + 1.47e3i)19-s + ⋯
L(s)  = 1  + (−0.731 − 1.26i)2-s + (0.822 − 1.42i)3-s + (−0.569 + 0.987i)4-s + (−0.257 − 0.445i)5-s − 2.40·6-s + 0.204·8-s + (−0.853 − 1.47i)9-s + (−0.376 + 0.651i)10-s + (0.336 − 0.582i)11-s + (0.937 + 1.62i)12-s + 0.493·13-s − 0.846·15-s + (0.420 + 0.728i)16-s + (−0.257 + 0.445i)17-s + (−1.24 + 2.16i)18-s + (0.540 + 0.936i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.524796 + 1.05864i\)
\(L(\frac12)\) \(\approx\) \(0.524796 + 1.05864i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (4.13 + 7.16i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (-12.8 + 22.2i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (14.3 + 24.8i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-135. + 233. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 300.T + 3.71e5T^{2} \)
17 \( 1 + (306. - 530. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-850. - 1.47e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.59e3 + 2.76e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 4.29e3T + 2.05e7T^{2} \)
31 \( 1 + (1.01e3 - 1.75e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.57e3 + 4.46e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 7.14e3T + 1.15e8T^{2} \)
43 \( 1 + 1.95e4T + 1.47e8T^{2} \)
47 \( 1 + (9.99e3 + 1.73e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.97e3 - 3.41e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.48e4 + 2.57e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.52e4 - 4.37e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.52e3 - 4.37e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.28e4T + 1.80e9T^{2} \)
73 \( 1 + (-5.55e3 + 9.62e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (4.09e4 + 7.09e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.18e5T + 3.93e9T^{2} \)
89 \( 1 + (-2.08e4 - 3.61e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 4.36e4T + 8.58e9T^{2} \)
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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

−13.46815660594953743718509733919, −12.43864853487351415298151395809, −11.78012883198974365528437504426, −10.26901664698926457758472585501, −8.651094269527156004196041444395, −8.270325144016314394138408746362, −6.45462510759515464325338775855, −3.43858696968138619582279864914, −1.92516815841393106644699527540, −0.72654830228491006091116332649, 3.30690619880750546809241072502, 4.99881473836371136679241936467, 6.84482566615479302024227384267, 8.144242333569784604442256469799, 9.217923958548189665202720973911, 9.966238153407530358170728604847, 11.44966649285710453252763021941, 13.76038377981734240884803036862, 14.78972955422460686070240966443, 15.49982006354446309800806852239

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