Properties

Label 2-7e2-49.8-c3-0-6
Degree $2$
Conductor $49$
Sign $0.992 + 0.120i$
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  + (0.777 + 0.974i)2-s + (−0.189 + 0.0912i)3-s + (1.43 − 6.28i)4-s + (7.66 − 3.69i)5-s + (−0.236 − 0.113i)6-s + (18.3 + 2.30i)7-s + (16.2 − 7.81i)8-s + (−16.8 + 21.0i)9-s + (9.55 + 4.60i)10-s + (2.58 + 3.24i)11-s + (0.301 + 1.32i)12-s + (−11.7 − 14.7i)13-s + (12.0 + 19.6i)14-s + (−1.11 + 1.40i)15-s + (−26.2 − 12.6i)16-s + (−4.87 − 21.3i)17-s + ⋯
L(s)  = 1  + (0.274 + 0.344i)2-s + (−0.0364 + 0.0175i)3-s + (0.179 − 0.785i)4-s + (0.685 − 0.330i)5-s + (−0.0160 − 0.00774i)6-s + (0.992 + 0.124i)7-s + (0.716 − 0.345i)8-s + (−0.622 + 0.780i)9-s + (0.302 + 0.145i)10-s + (0.0708 + 0.0888i)11-s + (0.00726 + 0.0318i)12-s + (−0.251 − 0.315i)13-s + (0.229 + 0.375i)14-s + (−0.0192 + 0.0241i)15-s + (−0.410 − 0.197i)16-s + (−0.0694 − 0.304i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ 0.992 + 0.120i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.77576 - 0.107430i\)
\(L(\frac12)\) \(\approx\) \(1.77576 - 0.107430i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-18.3 - 2.30i)T \)
good2 \( 1 + (-0.777 - 0.974i)T + (-1.78 + 7.79i)T^{2} \)
3 \( 1 + (0.189 - 0.0912i)T + (16.8 - 21.1i)T^{2} \)
5 \( 1 + (-7.66 + 3.69i)T + (77.9 - 97.7i)T^{2} \)
11 \( 1 + (-2.58 - 3.24i)T + (-296. + 1.29e3i)T^{2} \)
13 \( 1 + (11.7 + 14.7i)T + (-488. + 2.14e3i)T^{2} \)
17 \( 1 + (4.87 + 21.3i)T + (-4.42e3 + 2.13e3i)T^{2} \)
19 \( 1 + 46.4T + 6.85e3T^{2} \)
23 \( 1 + (18.3 - 80.4i)T + (-1.09e4 - 5.27e3i)T^{2} \)
29 \( 1 + (-44.6 - 195. i)T + (-2.19e4 + 1.05e4i)T^{2} \)
31 \( 1 + 276.T + 2.97e4T^{2} \)
37 \( 1 + (-48.3 - 211. i)T + (-4.56e4 + 2.19e4i)T^{2} \)
41 \( 1 + (-386. + 186. i)T + (4.29e4 - 5.38e4i)T^{2} \)
43 \( 1 + (-188. - 90.6i)T + (4.95e4 + 6.21e4i)T^{2} \)
47 \( 1 + (258. + 324. i)T + (-2.31e4 + 1.01e5i)T^{2} \)
53 \( 1 + (-39.2 + 172. i)T + (-1.34e5 - 6.45e4i)T^{2} \)
59 \( 1 + (-149. - 72.1i)T + (1.28e5 + 1.60e5i)T^{2} \)
61 \( 1 + (72.4 + 317. i)T + (-2.04e5 + 9.84e4i)T^{2} \)
67 \( 1 + 809.T + 3.00e5T^{2} \)
71 \( 1 + (-44.7 + 196. i)T + (-3.22e5 - 1.55e5i)T^{2} \)
73 \( 1 + (391. - 490. i)T + (-8.65e4 - 3.79e5i)T^{2} \)
79 \( 1 - 69.4T + 4.93e5T^{2} \)
83 \( 1 + (-441. + 553. i)T + (-1.27e5 - 5.57e5i)T^{2} \)
89 \( 1 + (-537. + 674. i)T + (-1.56e5 - 6.87e5i)T^{2} \)
97 \( 1 - 1.05e3T + 9.12e5T^{2} \)
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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

−14.84293423474865972901277353763, −14.15822790734302495443435687915, −13.11980803072007509895794054643, −11.40070225538004660739905820908, −10.45004551790079776373014526937, −9.038437119113619436330917933941, −7.50441407283186625446853366915, −5.74436871170022239840389270826, −4.94022568050661517363313352710, −1.84893996227795842442646283291, 2.33334868929224427395091330295, 4.20577787623093149001537846004, 6.12102826221404966808036347564, 7.72044003466933334126838454834, 9.021338551592049019626468491323, 10.72053389437073119308054892570, 11.66066911363262985136183694527, 12.73317948396254368454333845233, 14.04201395914871787303614372156, 14.77575474599426987437728614582

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