Properties

Label 2-7e2-49.2-c5-0-19
Degree $2$
Conductor $49$
Sign $-0.905 + 0.424i$
Analytic cond. $7.85880$
Root an. cond. $2.80335$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.16 + 0.666i)2-s + (6.26 − 15.9i)3-s + (−22.2 − 15.1i)4-s + (0.666 − 0.100i)5-s + (24.1 − 30.3i)6-s + (−129. + 6.24i)7-s + (−83.0 − 104. i)8-s + (−37.2 − 34.5i)9-s + (1.50 + 0.227i)10-s + (−298. + 276. i)11-s + (−380. + 259. i)12-s + (−104. − 459. i)13-s + (−283. − 72.8i)14-s + (2.57 − 11.2i)15-s + (204. + 520. i)16-s + (−74.3 − 992. i)17-s + ⋯
L(s)  = 1  + (0.381 + 0.117i)2-s + (0.401 − 1.02i)3-s + (−0.694 − 0.473i)4-s + (0.0119 − 0.00179i)5-s + (0.274 − 0.343i)6-s + (−0.998 + 0.0481i)7-s + (−0.458 − 0.575i)8-s + (−0.153 − 0.142i)9-s + (0.00476 + 0.000718i)10-s + (−0.743 + 0.690i)11-s + (−0.763 + 0.520i)12-s + (−0.171 − 0.753i)13-s + (−0.387 − 0.0992i)14-s + (0.00295 − 0.0129i)15-s + (0.199 + 0.508i)16-s + (−0.0624 − 0.832i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.905 + 0.424i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -0.905 + 0.424i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.243460 - 1.09297i\)
\(L(\frac12)\) \(\approx\) \(0.243460 - 1.09297i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (129. - 6.24i)T \)
good2 \( 1 + (-2.16 - 0.666i)T + (26.4 + 18.0i)T^{2} \)
3 \( 1 + (-6.26 + 15.9i)T + (-178. - 165. i)T^{2} \)
5 \( 1 + (-0.666 + 0.100i)T + (2.98e3 - 921. i)T^{2} \)
11 \( 1 + (298. - 276. i)T + (1.20e4 - 1.60e5i)T^{2} \)
13 \( 1 + (104. + 459. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (74.3 + 992. i)T + (-1.40e6 + 2.11e5i)T^{2} \)
19 \( 1 + (-904. + 1.56e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-317. + 4.24e3i)T + (-6.36e6 - 9.59e5i)T^{2} \)
29 \( 1 + (1.44e3 + 696. i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (-3.40e3 - 5.89e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-1.09e4 + 7.46e3i)T + (2.53e7 - 6.45e7i)T^{2} \)
41 \( 1 + (-2.09e3 - 2.63e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (2.53e3 - 3.17e3i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (1.27e4 + 3.92e3i)T + (1.89e8 + 1.29e8i)T^{2} \)
53 \( 1 + (-7.97e3 - 5.43e3i)T + (1.52e8 + 3.89e8i)T^{2} \)
59 \( 1 + (2.81e4 + 4.24e3i)T + (6.83e8 + 2.10e8i)T^{2} \)
61 \( 1 + (7.37e3 - 5.02e3i)T + (3.08e8 - 7.86e8i)T^{2} \)
67 \( 1 + (2.83e4 + 4.91e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-1.58e4 + 7.64e3i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (4.69e3 - 1.44e3i)T + (1.71e9 - 1.16e9i)T^{2} \)
79 \( 1 + (-1.72e3 + 2.98e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-1.28e4 + 5.63e4i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (-1.04e5 - 9.68e4i)T + (4.17e8 + 5.56e9i)T^{2} \)
97 \( 1 + 1.52e4T + 8.58e9T^{2} \)
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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

−13.78266195708488163452675139436, −13.09234945746863377063954215770, −12.40113848858728930780851366601, −10.32138230336709951097308005529, −9.233959269759965716060415404787, −7.67963147184603992511361018660, −6.45263332628918358969434409985, −4.87458660958280885821093013762, −2.74936785752997620840208373010, −0.48770933751949765395392779590, 3.22398118703256731168507657762, 4.14130662087865321187013670725, 5.79509264486812298756258813625, 7.954511711861638968933503987624, 9.314711716673291584381286782993, 10.01190875555463279985027653196, 11.69859341457566086675906052465, 13.04107524902594786741177706679, 13.83806024817021620306962699408, 15.09537351182298021302718130093

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