Properties

Label 2-7e2-49.43-c3-0-11
Degree $2$
Conductor $49$
Sign $-0.955 + 0.296i$
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  + (1.88 − 2.36i)2-s + (−5.83 − 2.81i)3-s + (−0.260 − 1.13i)4-s + (−12.6 − 6.09i)5-s + (−17.6 + 8.50i)6-s + (−3.94 − 18.0i)7-s + (18.6 + 8.97i)8-s + (9.31 + 11.6i)9-s + (−38.2 + 18.4i)10-s + (1.11 − 1.40i)11-s + (−1.68 + 7.38i)12-s + (19.6 − 24.6i)13-s + (−50.2 − 24.8i)14-s + (56.6 + 71.0i)15-s + (64.8 − 31.2i)16-s + (3.96 − 17.3i)17-s + ⋯
L(s)  = 1  + (0.667 − 0.837i)2-s + (−1.12 − 0.540i)3-s + (−0.0325 − 0.142i)4-s + (−1.13 − 0.544i)5-s + (−1.20 + 0.578i)6-s + (−0.212 − 0.977i)7-s + (0.823 + 0.396i)8-s + (0.345 + 0.432i)9-s + (−1.21 + 0.583i)10-s + (0.0306 − 0.0384i)11-s + (−0.0405 + 0.177i)12-s + (0.419 − 0.525i)13-s + (−0.959 − 0.474i)14-s + (0.975 + 1.22i)15-s + (1.01 − 0.488i)16-s + (0.0565 − 0.247i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.296i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.955 + 0.296i$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ -0.955 + 0.296i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.145740 - 0.962196i\)
\(L(\frac12)\) \(\approx\) \(0.145740 - 0.962196i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (3.94 + 18.0i)T \)
good2 \( 1 + (-1.88 + 2.36i)T + (-1.78 - 7.79i)T^{2} \)
3 \( 1 + (5.83 + 2.81i)T + (16.8 + 21.1i)T^{2} \)
5 \( 1 + (12.6 + 6.09i)T + (77.9 + 97.7i)T^{2} \)
11 \( 1 + (-1.11 + 1.40i)T + (-296. - 1.29e3i)T^{2} \)
13 \( 1 + (-19.6 + 24.6i)T + (-488. - 2.14e3i)T^{2} \)
17 \( 1 + (-3.96 + 17.3i)T + (-4.42e3 - 2.13e3i)T^{2} \)
19 \( 1 - 34.8T + 6.85e3T^{2} \)
23 \( 1 + (44.3 + 194. i)T + (-1.09e4 + 5.27e3i)T^{2} \)
29 \( 1 + (61.3 - 268. i)T + (-2.19e4 - 1.05e4i)T^{2} \)
31 \( 1 - 116.T + 2.97e4T^{2} \)
37 \( 1 + (-7.15 + 31.3i)T + (-4.56e4 - 2.19e4i)T^{2} \)
41 \( 1 + (294. + 141. i)T + (4.29e4 + 5.38e4i)T^{2} \)
43 \( 1 + (-405. + 195. i)T + (4.95e4 - 6.21e4i)T^{2} \)
47 \( 1 + (227. - 285. i)T + (-2.31e4 - 1.01e5i)T^{2} \)
53 \( 1 + (91.3 + 400. i)T + (-1.34e5 + 6.45e4i)T^{2} \)
59 \( 1 + (-753. + 362. i)T + (1.28e5 - 1.60e5i)T^{2} \)
61 \( 1 + (-1.60 + 7.03i)T + (-2.04e5 - 9.84e4i)T^{2} \)
67 \( 1 - 671.T + 3.00e5T^{2} \)
71 \( 1 + (159. + 699. i)T + (-3.22e5 + 1.55e5i)T^{2} \)
73 \( 1 + (-341. - 428. i)T + (-8.65e4 + 3.79e5i)T^{2} \)
79 \( 1 - 195.T + 4.93e5T^{2} \)
83 \( 1 + (-801. - 1.00e3i)T + (-1.27e5 + 5.57e5i)T^{2} \)
89 \( 1 + (378. + 474. i)T + (-1.56e5 + 6.87e5i)T^{2} \)
97 \( 1 - 665.T + 9.12e5T^{2} \)
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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

−14.20043683791754679431528162349, −12.84861920644446549030998567342, −12.34072261043912725198630574050, −11.34530476694917110731729148949, −10.56099652145871230109118189775, −8.163663568533916155584562340107, −6.90149556713670426348559776895, −4.98130365391546669877365438642, −3.66744826637582117436631560895, −0.72235578538241783759695500101, 3.99763202015455250463416825851, 5.42530592666347260142732592855, 6.39353243654275198420985999544, 7.83553167481656198112545061357, 9.865080963006929778449386677898, 11.31095234330484767796849777761, 11.85831188447278928628075499559, 13.53866199900298719503193924331, 14.98739385595416809656604687699, 15.64502110235931498852485810780

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