↑

Properties

Label 2-7e2-49.29-c1-0-2
Degree $2$
Conductor $49$
Sign $0.284 + 0.958i$
Analytic cond. $0.391266$
Root an. cond. $0.625513$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.240i)2-s + (−0.5 − 2.19i)3-s + (−1.05 − 1.32i)4-s + (0.321 + 1.40i)5-s + (−0.277 + 1.21i)6-s + (2.57 + 0.588i)7-s + (0.455 + 1.99i)8-s + (−1.84 + 0.888i)9-s + (0.178 − 0.781i)10-s + (3.32 + 1.60i)11-s + (−2.37 + 2.97i)12-s + (−5.25 − 2.52i)13-s + (−1.14 − 0.915i)14-s + (2.92 − 1.40i)15-s + (−0.500 + 2.19i)16-s + (−1.81 + 2.27i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.170i)2-s + (−0.288 − 1.26i)3-s + (−0.527 − 0.661i)4-s + (0.143 + 0.630i)5-s + (−0.113 + 0.496i)6-s + (0.974 + 0.222i)7-s + (0.161 + 0.706i)8-s + (−0.615 + 0.296i)9-s + (0.0564 − 0.247i)10-s + (1.00 + 0.482i)11-s + (−0.684 + 0.858i)12-s + (−1.45 − 0.701i)13-s + (−0.306 − 0.244i)14-s + (0.755 − 0.363i)15-s + (−0.125 + 0.547i)16-s + (−0.440 + 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.284 + 0.958i$
Analytic conductor: \(0.391266\)
Root analytic conductor: \(0.625513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :1/2),\ 0.284 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.518870 - 0.387242i\)
\(L(\frac12)\) \(\approx\) \(0.518870 - 0.387242i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.57 - 0.588i)T \)
good2 \( 1 + (0.5 + 0.240i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (0.5 + 2.19i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (-0.321 - 1.40i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (-3.32 - 1.60i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (5.25 + 2.52i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (1.81 - 2.27i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 - 1.93T + 19T^{2} \)
23 \( 1 + (-0.815 - 1.02i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (4.92 - 6.17i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + 3.24T + 31T^{2} \)
37 \( 1 + (1.13 - 1.42i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (-1.69 - 7.41i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.475 + 2.08i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (4.02 + 1.93i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (8.85 + 11.0i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-1.43 + 6.28i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (1.95 - 2.45i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 - 1.04T + 67T^{2} \)
71 \( 1 + (3.79 + 4.75i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-1.23 + 0.593i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + (-4.33 + 2.08i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-8.48 + 4.08i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + 1.35T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.77736602555407196915293336924, −14.53005280258572513130346407144, −13.05664830105896180208345831964, −11.92486805526406491332922905596, −10.79129952156238082617687851532, −9.427864679121662728757610216295, −7.88425090294222814812757439194, −6.68220390290657743294571309978, −5.11202791383763008523082064433, −1.78300378055637498511403651503, 4.14377745611663219865025426590, 5.00725578346383015569903243926, 7.42783279568733882555229530353, 9.003286936632787699039845243860, 9.549679581609570549357300247603, 11.14619287416916976111802017063, 12.22535641530999262982214203492, 13.80090108286178385872054575583, 14.85637591668963451958588552911, 16.26893870693630466811830928303

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