Properties

Label 2-7e2-49.2-c5-0-7
Degree $2$
Conductor $49$
Sign $0.332 + 0.943i$
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  + (−7.56 − 2.33i)2-s + (−5.53 + 14.0i)3-s + (25.3 + 17.2i)4-s + (−100. + 15.0i)5-s + (74.6 − 93.6i)6-s + (−91.8 + 91.5i)7-s + (6.78 + 8.50i)8-s + (10.1 + 9.42i)9-s + (792. + 119. i)10-s + (−221. + 205. i)11-s + (−383. + 261. i)12-s + (−175. − 767. i)13-s + (907. − 477. i)14-s + (341. − 1.49e3i)15-s + (−389. − 992. i)16-s + (53.0 + 708. i)17-s + ⋯
L(s)  = 1  + (−1.33 − 0.412i)2-s + (−0.354 + 0.903i)3-s + (0.790 + 0.539i)4-s + (−1.79 + 0.269i)5-s + (0.847 − 1.06i)6-s + (−0.708 + 0.705i)7-s + (0.0374 + 0.0469i)8-s + (0.0417 + 0.0387i)9-s + (2.50 + 0.377i)10-s + (−0.551 + 0.511i)11-s + (−0.767 + 0.523i)12-s + (−0.287 − 1.25i)13-s + (1.23 − 0.651i)14-s + (0.391 − 1.71i)15-s + (−0.380 − 0.969i)16-s + (0.0445 + 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.332 + 0.943i$
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: \((2,\ 49,\ (\ :5/2),\ 0.332 + 0.943i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0832246 - 0.0589368i\)
\(L(\frac12)\) \(\approx\) \(0.0832246 - 0.0589368i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (91.8 - 91.5i)T \)
good2 \( 1 + (7.56 + 2.33i)T + (26.4 + 18.0i)T^{2} \)
3 \( 1 + (5.53 - 14.0i)T + (-178. - 165. i)T^{2} \)
5 \( 1 + (100. - 15.0i)T + (2.98e3 - 921. i)T^{2} \)
11 \( 1 + (221. - 205. i)T + (1.20e4 - 1.60e5i)T^{2} \)
13 \( 1 + (175. + 767. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (-53.0 - 708. i)T + (-1.40e6 + 2.11e5i)T^{2} \)
19 \( 1 + (-713. + 1.23e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (89.7 - 1.19e3i)T + (-6.36e6 - 9.59e5i)T^{2} \)
29 \( 1 + (390. + 188. i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (-2.71e3 - 4.70e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-3.78e3 + 2.57e3i)T + (2.53e7 - 6.45e7i)T^{2} \)
41 \( 1 + (3.80e3 + 4.76e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (1.44e4 - 1.80e4i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (1.34e3 + 414. i)T + (1.89e8 + 1.29e8i)T^{2} \)
53 \( 1 + (-1.60e4 - 1.09e4i)T + (1.52e8 + 3.89e8i)T^{2} \)
59 \( 1 + (4.07e4 + 6.13e3i)T + (6.83e8 + 2.10e8i)T^{2} \)
61 \( 1 + (-4.27e4 + 2.91e4i)T + (3.08e8 - 7.86e8i)T^{2} \)
67 \( 1 + (3.60e3 + 6.23e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-4.11e4 + 1.98e4i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (2.35e4 - 7.26e3i)T + (1.71e9 - 1.16e9i)T^{2} \)
79 \( 1 + (-4.50e3 + 7.80e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-1.66e4 + 7.27e4i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (9.10e4 + 8.45e4i)T + (4.17e8 + 5.56e9i)T^{2} \)
97 \( 1 + 1.43e5T + 8.58e9T^{2} \)
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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

−15.21019044683836106496200304923, −12.69571986792945988261629819904, −11.54367914471233166722837774156, −10.62863178403994901688901094258, −9.754269999162464450927674936086, −8.356175788541134684338626587011, −7.39397318743355995695376416266, −4.92336225986220204648946160522, −3.13506486456294483684658202746, −0.13048253420679961909931122913, 0.821445905606465110224681048382, 4.00260525556600415505576007664, 6.75418605285498780177703779354, 7.43086484140279763974776000723, 8.374792934772267911285191148186, 9.836549123060366569761734376216, 11.32736601074886222295309016119, 12.25323430315577349972674468222, 13.49614089005738357923807902678, 15.38298552200850708843432830339

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