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Properties

Label 2-7e2-7.4-c3-0-5
Degree $2$
Conductor $49$
Sign $0.605 + 0.795i$
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  + (0.5 + 0.866i)2-s + (1 − 1.73i)3-s + (3.5 − 6.06i)4-s + (−8 − 13.8i)5-s + 1.99·6-s + 15·8-s + (11.5 + 19.9i)9-s + (7.99 − 13.8i)10-s + (4 − 6.92i)11-s + (−7 − 12.1i)12-s + 28·13-s − 31.9·15-s + (−20.5 − 35.5i)16-s + (−27 + 46.7i)17-s + (−11.5 + 19.9i)18-s + (55 + 95.2i)19-s + ⋯
L(s)  = 1  + (0.176 + 0.306i)2-s + (0.192 − 0.333i)3-s + (0.437 − 0.757i)4-s + (−0.715 − 1.23i)5-s + 0.136·6-s + 0.662·8-s + (0.425 + 0.737i)9-s + (0.252 − 0.438i)10-s + (0.109 − 0.189i)11-s + (−0.168 − 0.291i)12-s + 0.597·13-s − 0.550·15-s + (−0.320 − 0.554i)16-s + (−0.385 + 0.667i)17-s + (−0.150 + 0.260i)18-s + (0.664 + 1.15i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.42125 - 0.704553i\)
\(L(\frac12)\) \(\approx\) \(1.42125 - 0.704553i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-1 + 1.73i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (8 + 13.8i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-4 + 6.92i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 28T + 2.19e3T^{2} \)
17 \( 1 + (27 - 46.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-55 - 95.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (24 + 41.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 110T + 2.43e4T^{2} \)
31 \( 1 + (6 - 10.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-123 - 213. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 182T + 6.89e4T^{2} \)
43 \( 1 - 128T + 7.95e4T^{2} \)
47 \( 1 + (162 + 280. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-81 + 140. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (405 - 701. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-244 - 422. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (122 - 211. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 768T + 3.57e5T^{2} \)
73 \( 1 + (-351 + 607. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (220 + 381. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.30e3T + 5.71e5T^{2} \)
89 \( 1 + (365 + 632. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 294T + 9.12e5T^{2} \)
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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

−15.06345607608593443741083207213, −13.77265539156942069736713636512, −12.78562465886738234565250622269, −11.53097860870844103969127630530, −10.22456504513501829237877733088, −8.581594469756967746335107110100, −7.51040439803834874125895702703, −5.82706492510803844467686109859, −4.40179371253898927412225609898, −1.41194297365436200775023785024, 2.94662922883036392193526423579, 4.04952151809951250425555791784, 6.72818993492368257719414017129, 7.61513515356107126130760720471, 9.345898579344221918551748717528, 10.94955729157520126930059748839, 11.55427095132005709842121278934, 12.85498676468083956275921505827, 14.22546057348319997644493014164, 15.46889783696523621842747331906

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