Properties

Label 2-7e2-49.2-c5-0-1
Degree $2$
Conductor $49$
Sign $-0.889 + 0.456i$
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  + (4.24 + 1.31i)2-s + (−9.79 + 24.9i)3-s + (−10.1 − 6.89i)4-s + (−27.7 + 4.18i)5-s + (−74.3 + 93.1i)6-s + (129. − 8.58i)7-s + (−122. − 153. i)8-s + (−348. − 323. i)9-s + (−123. − 18.5i)10-s + (−488. + 453. i)11-s + (271. − 184. i)12-s + (−152. − 666. i)13-s + (560. + 132. i)14-s + (167. − 733. i)15-s + (−176. − 448. i)16-s + (46.2 + 616. i)17-s + ⋯
L(s)  = 1  + (0.750 + 0.231i)2-s + (−0.628 + 1.60i)3-s + (−0.316 − 0.215i)4-s + (−0.496 + 0.0747i)5-s + (−0.842 + 1.05i)6-s + (0.997 − 0.0662i)7-s + (−0.677 − 0.849i)8-s + (−1.43 − 1.33i)9-s + (−0.389 − 0.0587i)10-s + (−1.21 + 1.13i)11-s + (0.543 − 0.370i)12-s + (−0.249 − 1.09i)13-s + (0.764 + 0.181i)14-s + (0.192 − 0.841i)15-s + (−0.172 − 0.438i)16-s + (0.0387 + 0.517i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.889 + 0.456i$
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: \((2,\ 49,\ (\ :5/2),\ -0.889 + 0.456i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.150105 - 0.620734i\)
\(L(\frac12)\) \(\approx\) \(0.150105 - 0.620734i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-129. + 8.58i)T \)
good2 \( 1 + (-4.24 - 1.31i)T + (26.4 + 18.0i)T^{2} \)
3 \( 1 + (9.79 - 24.9i)T + (-178. - 165. i)T^{2} \)
5 \( 1 + (27.7 - 4.18i)T + (2.98e3 - 921. i)T^{2} \)
11 \( 1 + (488. - 453. i)T + (1.20e4 - 1.60e5i)T^{2} \)
13 \( 1 + (152. + 666. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (-46.2 - 616. i)T + (-1.40e6 + 2.11e5i)T^{2} \)
19 \( 1 + (125. - 216. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (320. - 4.27e3i)T + (-6.36e6 - 9.59e5i)T^{2} \)
29 \( 1 + (4.17e3 + 2.01e3i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (-3.78e3 - 6.55e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.14e3 + 3.51e3i)T + (2.53e7 - 6.45e7i)T^{2} \)
41 \( 1 + (-2.47e3 - 3.10e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (-1.54e3 + 1.94e3i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (2.07e4 + 6.40e3i)T + (1.89e8 + 1.29e8i)T^{2} \)
53 \( 1 + (1.38e4 + 9.41e3i)T + (1.52e8 + 3.89e8i)T^{2} \)
59 \( 1 + (-2.70e4 - 4.07e3i)T + (6.83e8 + 2.10e8i)T^{2} \)
61 \( 1 + (-407. + 277. i)T + (3.08e8 - 7.86e8i)T^{2} \)
67 \( 1 + (7.25e3 + 1.25e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-3.71e4 + 1.78e4i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (1.81e4 - 5.60e3i)T + (1.71e9 - 1.16e9i)T^{2} \)
79 \( 1 + (3.68e4 - 6.38e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (7.59e3 - 3.32e4i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (-3.59e4 - 3.33e4i)T + (4.17e8 + 5.56e9i)T^{2} \)
97 \( 1 - 1.67e5T + 8.58e9T^{2} \)
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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

−15.33896630521944579072207027049, −14.60348496823820025453801228903, −13.00951535159537058698428584619, −11.66264053219697656973384756297, −10.47330527897164512287617417313, −9.670433311079898105870255191714, −7.86824424859640683957918388415, −5.51684268098902050436435382433, −4.89674027517382379135103400105, −3.72421988598833871238993992882, 0.27295760513040118462927689042, 2.39620015879973194629319284924, 4.70135640511822791166490768866, 6.00049587164285532795579014421, 7.68232368904389507879189454546, 8.454972530962615023200062695880, 11.24486785686000086015964916702, 11.72811996287741280606409739425, 12.82719875054177285960308155042, 13.64105712240623889038638360032

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