Properties

Label 2-7e2-49.11-c3-0-9
Degree $2$
Conductor $49$
Sign $0.751 + 0.659i$
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  + (1.69 − 1.57i)2-s + (2.54 − 0.383i)3-s + (−0.199 + 2.66i)4-s + (4.20 − 10.7i)5-s + (3.70 − 4.64i)6-s + (16.1 − 9.13i)7-s + (15.3 + 19.2i)8-s + (−19.4 + 6.01i)9-s + (−9.69 − 24.7i)10-s + (−24.7 − 7.63i)11-s + (0.512 + 6.84i)12-s + (1.02 + 4.50i)13-s + (12.9 − 40.7i)14-s + (6.57 − 28.8i)15-s + (35.1 + 5.29i)16-s + (−75.7 + 51.6i)17-s + ⋯
L(s)  = 1  + (0.598 − 0.555i)2-s + (0.489 − 0.0737i)3-s + (−0.0249 + 0.332i)4-s + (0.375 − 0.957i)5-s + (0.251 − 0.315i)6-s + (0.869 − 0.493i)7-s + (0.678 + 0.851i)8-s + (−0.721 + 0.222i)9-s + (−0.306 − 0.781i)10-s + (−0.678 − 0.209i)11-s + (0.0123 + 0.164i)12-s + (0.0219 + 0.0960i)13-s + (0.246 − 0.778i)14-s + (0.113 − 0.496i)15-s + (0.548 + 0.0827i)16-s + (−1.08 + 0.736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ 0.751 + 0.659i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.98547 - 0.748074i\)
\(L(\frac12)\) \(\approx\) \(1.98547 - 0.748074i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-16.1 + 9.13i)T \)
good2 \( 1 + (-1.69 + 1.57i)T + (0.597 - 7.97i)T^{2} \)
3 \( 1 + (-2.54 + 0.383i)T + (25.8 - 7.95i)T^{2} \)
5 \( 1 + (-4.20 + 10.7i)T + (-91.6 - 85.0i)T^{2} \)
11 \( 1 + (24.7 + 7.63i)T + (1.09e3 + 749. i)T^{2} \)
13 \( 1 + (-1.02 - 4.50i)T + (-1.97e3 + 953. i)T^{2} \)
17 \( 1 + (75.7 - 51.6i)T + (1.79e3 - 4.57e3i)T^{2} \)
19 \( 1 + (-66.9 - 115. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (161. + 110. i)T + (4.44e3 + 1.13e4i)T^{2} \)
29 \( 1 + (-106. - 51.5i)T + (1.52e4 + 1.90e4i)T^{2} \)
31 \( 1 + (-62.4 + 108. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (19.2 + 256. i)T + (-5.00e4 + 7.54e3i)T^{2} \)
41 \( 1 + (-50.4 - 63.2i)T + (-1.53e4 + 6.71e4i)T^{2} \)
43 \( 1 + (61.5 - 77.1i)T + (-1.76e4 - 7.75e4i)T^{2} \)
47 \( 1 + (-370. + 344. i)T + (7.75e3 - 1.03e5i)T^{2} \)
53 \( 1 + (-37.4 + 500. i)T + (-1.47e5 - 2.21e4i)T^{2} \)
59 \( 1 + (-33.6 - 85.8i)T + (-1.50e5 + 1.39e5i)T^{2} \)
61 \( 1 + (27.8 + 372. i)T + (-2.24e5 + 3.38e4i)T^{2} \)
67 \( 1 + (350. - 607. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (384. - 185. i)T + (2.23e5 - 2.79e5i)T^{2} \)
73 \( 1 + (-587. - 545. i)T + (2.90e4 + 3.87e5i)T^{2} \)
79 \( 1 + (-207. - 359. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (4.75 - 20.8i)T + (-5.15e5 - 2.48e5i)T^{2} \)
89 \( 1 + (-395. + 122. i)T + (5.82e5 - 3.97e5i)T^{2} \)
97 \( 1 + 485.T + 9.12e5T^{2} \)
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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

−14.43466585682298009027410097697, −13.74461289868868215312601967896, −12.80769447939364761065692473982, −11.69427364980175113582095831376, −10.48077775891688177750519045004, −8.565962831146097002817828961313, −7.968727696067389741739221344563, −5.42853401841213771337123950454, −4.07987076982002080657763686278, −2.10817315908257502736576400229, 2.62852258414216789997756802388, 4.88051011132230004898816453429, 6.17657346987174295710637064367, 7.56325742829186946511486742486, 9.171107709203084210191814791072, 10.53184729399559633479696591218, 11.70374166265453622493476648696, 13.77552359505866645966026777751, 13.95270864509159997293712129201, 15.26385677603063917953067386359

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