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Properties

Label 2-7e2-7.4-c5-0-12
Degree $2$
Conductor $49$
Sign $-0.947 + 0.318i$
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  + (−1.40 − 2.43i)2-s + (3.27 − 5.67i)3-s + (12.0 − 20.8i)4-s + (−22.9 − 39.8i)5-s − 18.4·6-s − 157.·8-s + (100. + 173. i)9-s + (−64.7 + 112. i)10-s + (275. − 477. i)11-s + (−78.8 − 136. i)12-s − 1.09e3·13-s − 301.·15-s + (−163. − 282. i)16-s + (−590. + 1.02e3i)17-s + (281. − 487. i)18-s + (−583. − 1.00e3i)19-s + ⋯
L(s)  = 1  + (−0.248 − 0.430i)2-s + (0.210 − 0.363i)3-s + (0.376 − 0.651i)4-s + (−0.411 − 0.712i)5-s − 0.209·6-s − 0.872·8-s + (0.411 + 0.713i)9-s + (−0.204 + 0.354i)10-s + (0.687 − 1.19i)11-s + (−0.158 − 0.273i)12-s − 1.79·13-s − 0.345·15-s + (−0.159 − 0.275i)16-s + (−0.495 + 0.858i)17-s + (0.204 − 0.354i)18-s + (−0.370 − 0.641i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.947 + 0.318i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -0.947 + 0.318i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.198551 - 1.21404i\)
\(L(\frac12)\) \(\approx\) \(0.198551 - 1.21404i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (1.40 + 2.43i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (-3.27 + 5.67i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (22.9 + 39.8i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-275. + 477. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 1.09e3T + 3.71e5T^{2} \)
17 \( 1 + (590. - 1.02e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (583. + 1.00e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (22.1 + 38.4i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 3.32e3T + 2.05e7T^{2} \)
31 \( 1 + (-4.39e3 + 7.60e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-1.27e3 - 2.21e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.27e4T + 1.15e8T^{2} \)
43 \( 1 + 96.7T + 1.47e8T^{2} \)
47 \( 1 + (-3.83e3 - 6.65e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-5.97e3 + 1.03e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-4.92e3 + 8.53e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.92e4 + 3.33e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-3.37e4 + 5.84e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 6.13e4T + 1.80e9T^{2} \)
73 \( 1 + (-925. + 1.60e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-4.26 - 7.38i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 9.50e4T + 3.93e9T^{2} \)
89 \( 1 + (-2.68e4 - 4.64e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 3.11e3T + 8.58e9T^{2} \)
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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

−14.07972758303177963371686791684, −12.75157770724582394012763628571, −11.71509202087348875847094919792, −10.56967321572820537385970211587, −9.267702657198672231355161682330, −8.001246975133458463385870135317, −6.42138441784189415185922489682, −4.70268449039605136858083475364, −2.35230236436993423393757883629, −0.65574159389265924212460772692, 2.75957390372344750956776190908, 4.34602994532188341267153832990, 6.81213456497124376439808225837, 7.36760983071054787822316805893, 9.067197218529920161270882856196, 10.15741176881194643927593844436, 11.87739039274612842804778901686, 12.43764926345742456925846337275, 14.54761112449404074785941363194, 15.08109552370590043288882914781

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