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Properties

Label 2-7e2-7.2-c3-0-3
Degree $2$
Conductor $49$
Sign $0.947 + 0.318i$
Analytic cond. $2.89109$
Root an. cond. $1.70032$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.26 + 3.92i)2-s + (−1.78 − 3.09i)3-s + (−6.26 − 10.8i)4-s + (6.73 − 11.6i)5-s + 16.2·6-s + 20.5·8-s + (7.09 − 12.2i)9-s + (30.5 + 52.8i)10-s + (−0.406 − 0.704i)11-s + (−22.4 + 38.8i)12-s + 34.9·13-s − 48.2·15-s + (3.60 − 6.25i)16-s + (−58.8 − 101. i)17-s + (32.1 + 55.6i)18-s + (46.6 − 80.7i)19-s + ⋯
L(s)  = 1  + (−0.800 + 1.38i)2-s + (−0.344 − 0.596i)3-s + (−0.783 − 1.35i)4-s + (0.602 − 1.04i)5-s + 1.10·6-s + 0.907·8-s + (0.262 − 0.455i)9-s + (0.965 + 1.67i)10-s + (−0.0111 − 0.0193i)11-s + (−0.539 + 0.934i)12-s + 0.745·13-s − 0.830·15-s + (0.0564 − 0.0976i)16-s + (−0.839 − 1.45i)17-s + (0.420 + 0.728i)18-s + (0.563 − 0.975i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.947 + 0.318i$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ 0.947 + 0.318i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.782722 - 0.128011i\)
\(L(\frac12)\) \(\approx\) \(0.782722 - 0.128011i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.26 - 3.92i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (1.78 + 3.09i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-6.73 + 11.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (0.406 + 0.704i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 34.9T + 2.19e3T^{2} \)
17 \( 1 + (58.8 + 101. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-46.6 + 80.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (60.1 - 104. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 8.56T + 2.43e4T^{2} \)
31 \( 1 + (-41.0 - 71.1i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (14.4 - 24.9i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 70.5T + 6.89e4T^{2} \)
43 \( 1 - 417.T + 7.95e4T^{2} \)
47 \( 1 + (169. - 292. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (74.5 + 129. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-47.0 - 81.5i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-60.2 + 104. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-396. - 686. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 449.T + 3.57e5T^{2} \)
73 \( 1 + (-234. - 406. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-509. + 883. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 104.T + 5.71e5T^{2} \)
89 \( 1 + (-786. + 1.36e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 550.T + 9.12e5T^{2} \)
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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

−15.64181403463783569024987388884, −13.95690059111032710720061594872, −13.04635243551275214031838277420, −11.61948052517167346324257474739, −9.552766365260696283181298888959, −8.913304030631740610189745697626, −7.44353828441020756554370618209, −6.34950089040177703848099872359, −5.14311828032920619167830366237, −0.873325273054187424694295672963, 2.07779906582429401816152577618, 3.85175445141334880018169964442, 6.17096336750037277830043142027, 8.244737092293434142992846587696, 9.745761829158069258413976589985, 10.56830196236854601087981156858, 11.04206908702390124839347566964, 12.52047146997637901350518944446, 13.80443978682955798038588642503, 15.21471285477266461014359389412

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