Properties

Label 2-7e2-49.25-c5-0-9
Degree $2$
Conductor $49$
Sign $-0.615 - 0.788i$
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  + (−1.52 + 0.471i)2-s + (10.3 + 26.3i)3-s + (−24.3 + 16.5i)4-s + (90.2 + 13.6i)5-s + (−28.2 − 35.3i)6-s + (124. + 36.1i)7-s + (61.3 − 76.8i)8-s + (−407. + 378. i)9-s + (−144. + 21.7i)10-s + (−127. − 118. i)11-s + (−687. − 468. i)12-s + (67.7 − 296. i)13-s + (−207. + 3.37i)14-s + (574. + 2.51e3i)15-s + (286. − 730. i)16-s + (−62.2 + 831. i)17-s + ⋯
L(s)  = 1  + (−0.270 + 0.0834i)2-s + (0.662 + 1.68i)3-s + (−0.760 + 0.518i)4-s + (1.61 + 0.243i)5-s + (−0.319 − 0.401i)6-s + (0.960 + 0.279i)7-s + (0.338 − 0.424i)8-s + (−1.67 + 1.55i)9-s + (−0.457 + 0.0688i)10-s + (−0.318 − 0.295i)11-s + (−1.37 − 0.939i)12-s + (0.111 − 0.486i)13-s + (−0.282 + 0.00460i)14-s + (0.659 + 2.88i)15-s + (0.279 − 0.713i)16-s + (−0.0522 + 0.697i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.615 - 0.788i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -0.615 - 0.788i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.868995 + 1.78120i\)
\(L(\frac12)\) \(\approx\) \(0.868995 + 1.78120i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-124. - 36.1i)T \)
good2 \( 1 + (1.52 - 0.471i)T + (26.4 - 18.0i)T^{2} \)
3 \( 1 + (-10.3 - 26.3i)T + (-178. + 165. i)T^{2} \)
5 \( 1 + (-90.2 - 13.6i)T + (2.98e3 + 921. i)T^{2} \)
11 \( 1 + (127. + 118. i)T + (1.20e4 + 1.60e5i)T^{2} \)
13 \( 1 + (-67.7 + 296. i)T + (-3.34e5 - 1.61e5i)T^{2} \)
17 \( 1 + (62.2 - 831. i)T + (-1.40e6 - 2.11e5i)T^{2} \)
19 \( 1 + (912. + 1.57e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (25.5 + 341. i)T + (-6.36e6 + 9.59e5i)T^{2} \)
29 \( 1 + (-3.32e3 + 1.59e3i)T + (1.27e7 - 1.60e7i)T^{2} \)
31 \( 1 + (-2.76e3 + 4.78e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (1.08e4 + 7.39e3i)T + (2.53e7 + 6.45e7i)T^{2} \)
41 \( 1 + (-4.56e3 + 5.72e3i)T + (-2.57e7 - 1.12e8i)T^{2} \)
43 \( 1 + (-1.45e4 - 1.82e4i)T + (-3.27e7 + 1.43e8i)T^{2} \)
47 \( 1 + (1.94e4 - 5.98e3i)T + (1.89e8 - 1.29e8i)T^{2} \)
53 \( 1 + (-3.22e3 + 2.19e3i)T + (1.52e8 - 3.89e8i)T^{2} \)
59 \( 1 + (-2.04e4 + 3.07e3i)T + (6.83e8 - 2.10e8i)T^{2} \)
61 \( 1 + (-1.41e4 - 9.66e3i)T + (3.08e8 + 7.86e8i)T^{2} \)
67 \( 1 + (3.37e4 - 5.84e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (2.00e4 + 9.65e3i)T + (1.12e9 + 1.41e9i)T^{2} \)
73 \( 1 + (-4.14e4 - 1.27e4i)T + (1.71e9 + 1.16e9i)T^{2} \)
79 \( 1 + (3.31e3 + 5.73e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (5.68e3 + 2.49e4i)T + (-3.54e9 + 1.70e9i)T^{2} \)
89 \( 1 + (-4.33e4 + 4.01e4i)T + (4.17e8 - 5.56e9i)T^{2} \)
97 \( 1 + 2.53e3T + 8.58e9T^{2} \)
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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

−14.85798769153059346782104878824, −14.08502568741223967635402497674, −13.15616566154207371500421579934, −10.88245791768769366119810910403, −10.02620725678823402286299419785, −9.064121452619467081108018631308, −8.266761318863277201672748764573, −5.55877296370753584345981514071, −4.42119243225216506057893229623, −2.64493763566687646096164982483, 1.22558514101056322002789025670, 2.04850313365904570020978470056, 5.23236134624547338218401533511, 6.60938550095410020054496118823, 8.166814537144309676836264949067, 9.073165417500715038103811120509, 10.37756968446964161669628291233, 12.23422073280486043785711934093, 13.43086170137104442799287456587, 13.94150645519481074563005475504

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