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Properties

Label 2-7e2-49.29-c1-0-0
Degree $2$
Conductor $49$
Sign $0.802 - 0.596i$
Analytic cond. $0.391266$
Root an. cond. $0.625513$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.78 − 0.858i)2-s + (0.590 + 2.58i)3-s + (1.19 + 1.49i)4-s + (0.359 + 1.57i)5-s + (1.16 − 5.11i)6-s + (1.95 − 1.77i)7-s + (0.0391 + 0.171i)8-s + (−3.63 + 1.75i)9-s + (0.710 − 3.11i)10-s + (−3.98 − 1.92i)11-s + (−3.16 + 3.96i)12-s + (1.03 + 0.498i)13-s + (−5.01 + 1.48i)14-s + (−3.86 + 1.85i)15-s + (0.927 − 4.06i)16-s + (4.05 − 5.07i)17-s + ⋯
L(s)  = 1  + (−1.25 − 0.606i)2-s + (0.340 + 1.49i)3-s + (0.595 + 0.747i)4-s + (0.160 + 0.704i)5-s + (0.476 − 2.08i)6-s + (0.740 − 0.671i)7-s + (0.0138 + 0.0605i)8-s + (−1.21 + 0.583i)9-s + (0.224 − 0.985i)10-s + (−1.20 − 0.579i)11-s + (−0.912 + 1.14i)12-s + (0.287 + 0.138i)13-s + (−1.34 + 0.396i)14-s + (−0.996 + 0.480i)15-s + (0.231 − 1.01i)16-s + (0.982 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.802 - 0.596i$
Analytic conductor: \(0.391266\)
Root analytic conductor: \(0.625513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :1/2),\ 0.802 - 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.508364 + 0.168371i\)
\(L(\frac12)\) \(\approx\) \(0.508364 + 0.168371i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.95 + 1.77i)T \)
good2 \( 1 + (1.78 + 0.858i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (-0.590 - 2.58i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (-0.359 - 1.57i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (3.98 + 1.92i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.03 - 0.498i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-4.05 + 5.07i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + 1.87T + 19T^{2} \)
23 \( 1 + (0.184 + 0.231i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-1.70 + 2.13i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + 5.93T + 31T^{2} \)
37 \( 1 + (5.66 - 7.10i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (-0.655 - 2.87i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (0.200 - 0.878i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-1.53 - 0.738i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (1.35 + 1.69i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (0.611 - 2.67i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (2.01 - 2.52i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + 3.33T + 67T^{2} \)
71 \( 1 + (7.37 + 9.25i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-5.15 + 2.48i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + (6.48 - 3.12i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-12.2 + 5.90i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 - 3.26T + 97T^{2} \)
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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

−16.04637467595207949913499602759, −14.72218116659349082661123887864, −13.85712214538973916985139203567, −11.44433094187522623664375828890, −10.56016828210391898540885204846, −10.11941678249469289590976424478, −8.829035128440735504168972347254, −7.68468436331260203568446413601, −5.02164354716780403878355856088, −3.02981683616714777670576781868, 1.70047552364222161280466687435, 5.67849383471930044347932682662, 7.30440904455159800279083360767, 8.152515937866388113165549333238, 8.863702849325000007008059295286, 10.55080814553694573942547865908, 12.47680713468380208366893329638, 12.92626898882112211562197717113, 14.54295574769840226888502158462, 15.74393180931112803905068547587

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