Properties

Label 2-7e2-7.4-c5-0-2
Degree $2$
Conductor $49$
Sign $-0.701 + 0.712i$
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  + (5 + 8.66i)2-s + (−7 + 12.1i)3-s + (−34.0 + 58.8i)4-s + (−28 − 48.4i)5-s − 140·6-s − 360.·8-s + (23.5 + 40.7i)9-s + (280. − 484. i)10-s + (−116 + 200. i)11-s + (−476 − 824. i)12-s + 140·13-s + 784·15-s + (−712. − 1.23e3i)16-s + (−861 + 1.49e3i)17-s + (−235. + 407. i)18-s + (−49 − 84.8i)19-s + ⋯
L(s)  = 1  + (0.883 + 1.53i)2-s + (−0.449 + 0.777i)3-s + (−1.06 + 1.84i)4-s + (−0.500 − 0.867i)5-s − 1.58·6-s − 1.98·8-s + (0.0967 + 0.167i)9-s + (0.885 − 1.53i)10-s + (−0.289 + 0.500i)11-s + (−0.954 − 1.65i)12-s + 0.229·13-s + 0.899·15-s + (−0.695 − 1.20i)16-s + (−0.722 + 1.25i)17-s + (−0.170 + 0.296i)18-s + (−0.0311 − 0.0539i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.701 + 0.712i$
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: \((2,\ 49,\ (\ :5/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.578060 - 1.37934i\)
\(L(\frac12)\) \(\approx\) \(0.578060 - 1.37934i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-5 - 8.66i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (7 - 12.1i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (28 + 48.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (116 - 200. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 140T + 3.71e5T^{2} \)
17 \( 1 + (861 - 1.49e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (49 + 84.8i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (912 + 1.57e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 3.41e3T + 2.05e7T^{2} \)
31 \( 1 + (3.82e3 - 6.61e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-5.19e3 - 9.00e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.79e4T + 1.15e8T^{2} \)
43 \( 1 - 1.08e4T + 1.47e8T^{2} \)
47 \( 1 + (-4.66e3 - 8.07e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.13e3 - 1.95e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.36e3 - 2.36e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.28e4 - 2.22e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.42e4 + 4.19e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 5.85e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.40e4 + 5.89e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (1.58e4 + 2.75e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 2.05e4T + 3.93e9T^{2} \)
89 \( 1 + (2.52e4 + 4.38e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 5.85e4T + 8.58e9T^{2} \)
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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

−15.59964226317481342267609771744, −14.51953614573807688153170664752, −13.13838685224223409936876015023, −12.35743687166272342580647322623, −10.61744545586446670896712677784, −8.814950747871708736838933490252, −7.74602683652180958488176801987, −6.17941847040448502626987433539, −4.83201982646118943624290197710, −4.18272332541666713329523285204, 0.62543727073202147139343221438, 2.51225456828202737331967992047, 3.95185726408973716945506684858, 5.78738669456850938922649231359, 7.29948338626288867001525896968, 9.524240352591819466082394084054, 11.04988881648716627805627173981, 11.45177766655167248021121568451, 12.59492164439596816431039260967, 13.50198593976429974427420062163

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