Properties

Label 2-7e2-49.2-c5-0-5
Degree $2$
Conductor $49$
Sign $0.245 - 0.969i$
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  + (−1.11 − 0.343i)2-s + (1.70 − 4.33i)3-s + (−25.3 − 17.2i)4-s + (−34.5 + 5.20i)5-s + (−3.38 + 4.24i)6-s + (61.0 + 114. i)7-s + (45.5 + 57.1i)8-s + (162. + 150. i)9-s + (40.3 + 6.07i)10-s + (−26.0 + 24.1i)11-s + (−117. + 80.3i)12-s + (43.5 + 190. i)13-s + (−28.7 − 148. i)14-s + (−36.1 + 158. i)15-s + (327. + 833. i)16-s + (43.0 + 575. i)17-s + ⋯
L(s)  = 1  + (−0.197 − 0.0607i)2-s + (0.109 − 0.277i)3-s + (−0.791 − 0.539i)4-s + (−0.618 + 0.0931i)5-s + (−0.0383 + 0.0481i)6-s + (0.470 + 0.882i)7-s + (0.251 + 0.315i)8-s + (0.667 + 0.619i)9-s + (0.127 + 0.0192i)10-s + (−0.0648 + 0.0601i)11-s + (−0.236 + 0.161i)12-s + (0.0714 + 0.313i)13-s + (−0.0391 − 0.202i)14-s + (−0.0415 + 0.181i)15-s + (0.319 + 0.813i)16-s + (0.0361 + 0.482i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.245 - 0.969i$
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: \((2,\ 49,\ (\ :5/2),\ 0.245 - 0.969i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.728973 + 0.567329i\)
\(L(\frac12)\) \(\approx\) \(0.728973 + 0.567329i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-61.0 - 114. i)T \)
good2 \( 1 + (1.11 + 0.343i)T + (26.4 + 18.0i)T^{2} \)
3 \( 1 + (-1.70 + 4.33i)T + (-178. - 165. i)T^{2} \)
5 \( 1 + (34.5 - 5.20i)T + (2.98e3 - 921. i)T^{2} \)
11 \( 1 + (26.0 - 24.1i)T + (1.20e4 - 1.60e5i)T^{2} \)
13 \( 1 + (-43.5 - 190. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (-43.0 - 575. i)T + (-1.40e6 + 2.11e5i)T^{2} \)
19 \( 1 + (534. - 926. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (140. - 1.86e3i)T + (-6.36e6 - 9.59e5i)T^{2} \)
29 \( 1 + (1.55e3 + 749. i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (864. + 1.49e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (2.63e3 - 1.79e3i)T + (2.53e7 - 6.45e7i)T^{2} \)
41 \( 1 + (-2.25e3 - 2.82e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (-2.97e3 + 3.73e3i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (-2.36e3 - 730. i)T + (1.89e8 + 1.29e8i)T^{2} \)
53 \( 1 + (-1.76e4 - 1.20e4i)T + (1.52e8 + 3.89e8i)T^{2} \)
59 \( 1 + (1.67e4 + 2.52e3i)T + (6.83e8 + 2.10e8i)T^{2} \)
61 \( 1 + (1.61e4 - 1.10e4i)T + (3.08e8 - 7.86e8i)T^{2} \)
67 \( 1 + (-3.15e4 - 5.46e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-2.26e4 + 1.08e4i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (2.45e4 - 7.55e3i)T + (1.71e9 - 1.16e9i)T^{2} \)
79 \( 1 + (3.33e4 - 5.77e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-1.76e4 + 7.71e4i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (8.09e4 + 7.50e4i)T + (4.17e8 + 5.56e9i)T^{2} \)
97 \( 1 - 1.03e5T + 8.58e9T^{2} \)
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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

−14.85533706044512710481387516756, −13.72293730636220684307065172888, −12.58536370676622100243061318548, −11.29516846712010255157483989118, −10.00748396575658008477927291379, −8.684199981206236255131865038732, −7.65500291792389851133395356339, −5.68461791817782245362836704375, −4.24511465521489888650298234675, −1.72560609480247330122206608677, 0.54358141668527003877034500180, 3.68938247161593873238020425265, 4.67216640865017056115965356552, 7.11466203557820609281912340858, 8.189115808610760155457611141640, 9.424931726866740066268948371707, 10.67761451130883274863332839543, 12.14217089340697503288783700344, 13.20911382215511964662039325943, 14.33010428573721234043478558240

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