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Properties

Label 2-7e2-49.15-c1-0-0
Degree $2$
Conductor $49$
Sign $-0.462 - 0.886i$
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  + (−0.5 + 2.19i)2-s + (−0.5 + 0.626i)3-s + (−2.74 − 1.32i)4-s + (0.153 − 0.193i)5-s + (−1.12 − 1.40i)6-s + (2.06 − 1.64i)7-s + (1.46 − 1.84i)8-s + (0.524 + 2.29i)9-s + (0.346 + 0.433i)10-s + (−0.233 + 1.02i)11-s + (2.20 − 1.06i)12-s + (1.18 − 5.21i)13-s + (2.57 + 5.35i)14-s + (0.0440 + 0.193i)15-s + (−0.499 − 0.626i)16-s + (4.44 − 2.14i)17-s + ⋯
L(s)  = 1  + (−0.353 + 1.54i)2-s + (−0.288 + 0.361i)3-s + (−1.37 − 0.661i)4-s + (0.0688 − 0.0863i)5-s + (−0.458 − 0.575i)6-s + (0.781 − 0.623i)7-s + (0.519 − 0.651i)8-s + (0.174 + 0.765i)9-s + (0.109 + 0.137i)10-s + (−0.0703 + 0.308i)11-s + (0.635 − 0.306i)12-s + (0.329 − 1.44i)13-s + (0.689 + 1.43i)14-s + (0.0113 + 0.0498i)15-s + (−0.124 − 0.156i)16-s + (1.07 − 0.519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.462 - 0.886i$
Analytic conductor: \(0.391266\)
Root analytic conductor: \(0.625513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :1/2),\ -0.462 - 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.344530 + 0.568338i\)
\(L(\frac12)\) \(\approx\) \(0.344530 + 0.568338i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.06 + 1.64i)T \)
good2 \( 1 + (0.5 - 2.19i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (0.5 - 0.626i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (-0.153 + 0.193i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (0.233 - 1.02i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-1.18 + 5.21i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-4.44 + 2.14i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + 5.85T + 19T^{2} \)
23 \( 1 + (5.44 + 2.62i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (2.04 - 0.984i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + 0.198T + 31T^{2} \)
37 \( 1 + (-5.29 + 2.55i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (3.04 - 3.82i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-2.67 - 3.35i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (1.82 - 7.98i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (3.08 + 1.48i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (2.96 + 3.71i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (2.96 - 1.43i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + 3.35T + 67T^{2} \)
71 \( 1 + (-3.65 - 1.76i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-3.09 - 13.5i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 - 8.64T + 79T^{2} \)
83 \( 1 + (2.62 + 11.4i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (3.70 + 16.2i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + 1.69T + 97T^{2} \)
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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

−16.14729968126433965437583585362, −15.07884536754128687808074944444, −14.23222047207391033359426888504, −12.95542524804237484982919967516, −11.01768686529765957570055858559, −9.927321710363870396925145105487, −8.204557174358981147082970581407, −7.54870536067996224141133890333, −5.82627508281190963090632170647, −4.69936541159008727976790338659, 1.86629264847093972939950744102, 4.00265163688008572574436800220, 6.23773977415016489937196193766, 8.365718240876043026805650426249, 9.490232331102243072927302182147, 10.79705427716430034829504805100, 11.86478435677985334861493276968, 12.35093847829526929884364388089, 13.81067500718346598040186783447, 15.08262284830239736912189077817

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