Properties

Label 2-7e2-7.2-c9-0-26
Degree $2$
Conductor $49$
Sign $-0.701 - 0.712i$
Analytic cond. $25.2367$
Root an. cond. $5.02361$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (17.0 − 29.5i)2-s + (−39.8 − 68.9i)3-s + (−327. − 567. i)4-s + (711. − 1.23e3i)5-s − 2.72e3·6-s − 4.88e3·8-s + (6.66e3 − 1.15e4i)9-s + (−2.43e4 − 4.21e4i)10-s + (−3.46e4 − 6.00e4i)11-s + (−2.60e4 + 4.51e4i)12-s − 1.05e5·13-s − 1.13e5·15-s + (8.42e4 − 1.45e5i)16-s + (2.84e5 + 4.92e5i)17-s + (−2.27e5 − 3.94e5i)18-s + (−1.98e5 + 3.43e5i)19-s + ⋯
L(s)  = 1  + (0.754 − 1.30i)2-s + (−0.283 − 0.491i)3-s + (−0.639 − 1.10i)4-s + (0.509 − 0.882i)5-s − 0.857·6-s − 0.421·8-s + (0.338 − 0.586i)9-s + (−0.769 − 1.33i)10-s + (−0.714 − 1.23i)11-s + (−0.363 + 0.629i)12-s − 1.02·13-s − 0.578·15-s + (0.321 − 0.556i)16-s + (0.825 + 1.42i)17-s + (−0.511 − 0.886i)18-s + (−0.348 + 0.604i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(25.2367\)
Root analytic conductor: \(5.02361\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :9/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.948899 + 2.26422i\)
\(L(\frac12)\) \(\approx\) \(0.948899 + 2.26422i\)
\(L(\frac{11}{2})\) 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 \( 1 + (-17.0 + 29.5i)T + (-256 - 443. i)T^{2} \)
3 \( 1 + (39.8 + 68.9i)T + (-9.84e3 + 1.70e4i)T^{2} \)
5 \( 1 + (-711. + 1.23e3i)T + (-9.76e5 - 1.69e6i)T^{2} \)
11 \( 1 + (3.46e4 + 6.00e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 + 1.05e5T + 1.06e10T^{2} \)
17 \( 1 + (-2.84e5 - 4.92e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (1.98e5 - 3.43e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (-3.10e5 + 5.37e5i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 - 4.87e6T + 1.45e13T^{2} \)
31 \( 1 + (7.12e5 + 1.23e6i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 + (6.55e6 - 1.13e7i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 - 2.03e7T + 3.27e14T^{2} \)
43 \( 1 + 1.11e7T + 5.02e14T^{2} \)
47 \( 1 + (9.96e6 - 1.72e7i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (2.82e7 + 4.89e7i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (5.46e7 + 9.46e7i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (-1.60e7 + 2.77e7i)T + (-5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (4.01e7 + 6.95e7i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 - 2.07e8T + 4.58e16T^{2} \)
73 \( 1 + (1.35e8 + 2.34e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (-2.58e8 + 4.47e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 - 6.82e8T + 1.86e17T^{2} \)
89 \( 1 + (7.35e7 - 1.27e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 + 1.09e9T + 7.60e17T^{2} \)
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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

−12.64288278688217395991011027293, −12.20112555064956523639972932813, −10.76685982459509956918754421071, −9.738639050276298173425762856615, −8.123622890357701139243475218477, −6.06173267268544657392496246570, −4.86950226508123662015197375009, −3.32228398991602700695907387039, −1.72562283228458555323329879028, −0.65900477991308425433185087151, 2.52985651739389545735380401517, 4.61902734586773849304609477260, 5.34363675675330562293275924321, 6.94263716305005271820876599867, 7.56466152201370973350790106577, 9.748161149240107302190641279121, 10.63071155193702943467907703303, 12.41235505351247899245959418193, 13.66885278885964119191645166387, 14.51639756074186874833496006183

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