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Properties

Label 2-7e2-7.4-c5-0-1
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  + (−0.362 − 0.627i)2-s + (−9.82 + 17.0i)3-s + (15.7 − 27.2i)4-s + (23.3 + 40.4i)5-s + 14.2·6-s − 46.0·8-s + (−71.5 − 123. i)9-s + (16.9 − 29.3i)10-s + (−333. + 576. i)11-s + (309. + 535. i)12-s − 650.·13-s − 918.·15-s + (−486. − 843. i)16-s + (−593. + 1.02e3i)17-s + (−51.8 + 89.8i)18-s + (782. + 1.35e3i)19-s + ⋯
L(s)  = 1  + (−0.0640 − 0.111i)2-s + (−0.630 + 1.09i)3-s + (0.491 − 0.851i)4-s + (0.418 + 0.724i)5-s + 0.161·6-s − 0.254·8-s + (−0.294 − 0.510i)9-s + (0.0535 − 0.0928i)10-s + (−0.829 + 1.43i)11-s + (0.619 + 1.07i)12-s − 1.06·13-s − 1.05·15-s + (−0.475 − 0.823i)16-s + (−0.498 + 0.862i)17-s + (−0.0377 + 0.0653i)18-s + (0.497 + 0.861i)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

\(L(3)\) \(\approx\) \(0.473124 + 0.954404i\)
\(L(\frac12)\) \(\approx\) \(0.473124 + 0.954404i\)
\(L(\frac{7}{2})\) 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 \( 1 + (0.362 + 0.627i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (9.82 - 17.0i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-23.3 - 40.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (333. - 576. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 650.T + 3.71e5T^{2} \)
17 \( 1 + (593. - 1.02e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-782. - 1.35e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-550. - 952. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 2.39e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.02e3 + 1.77e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (538. + 933. i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.09e3T + 1.15e8T^{2} \)
43 \( 1 - 1.65e4T + 1.47e8T^{2} \)
47 \( 1 + (-4.14e3 - 7.18e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (2.75e3 - 4.77e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-7.11e3 + 1.23e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-7.11e3 - 1.23e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (9.86e3 - 1.70e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.45e4T + 1.80e9T^{2} \)
73 \( 1 + (1.42e4 - 2.47e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.53e4 - 2.65e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 675.T + 3.93e9T^{2} \)
89 \( 1 + (6.29e4 + 1.09e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 2.29e4T + 8.58e9T^{2} \)
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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

−15.15772974296901709761801789716, −14.30600626420830249472570589770, −12.44586884867239820162807929056, −11.05969591557011370712827270491, −10.16397236139366064825057714554, −9.785581539444798938856820594136, −7.27394421430890433473679652836, −5.82839495383027326556469513760, −4.65935774341676443746581319712, −2.28808239377355775953340333668, 0.57190253216914595809808973200, 2.64126999312186990840447280197, 5.24738485581897909930788037400, 6.67656425708448539525260830671, 7.74115866419855444260753250448, 9.035971162101033416462318401295, 11.05540164358768399452771448413, 12.04000239538679946351126282871, 12.92938265212926295936264975007, 13.69519059095220160826140256393

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