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Properties

Label 2-7e2-7.5-c4-0-9
Degree $2$
Conductor $49$
Sign $-0.580 + 0.814i$
Analytic cond. $5.06512$
Root an. cond. $2.25058$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.49 − 6.05i)2-s + (6.83 − 3.94i)3-s + (−16.4 − 28.4i)4-s + (19.0 + 11.0i)5-s − 55.1i·6-s − 117.·8-s + (−9.36 + 16.2i)9-s + (133. − 76.9i)10-s + (−22.3 − 38.7i)11-s + (−224. − 129. i)12-s + 311. i·13-s + 173.·15-s + (−149. + 258. i)16-s + (61.9 − 35.7i)17-s + (65.4 + 113. i)18-s + (−241. − 139. i)19-s + ⋯
L(s)  = 1  + (0.873 − 1.51i)2-s + (0.759 − 0.438i)3-s + (−1.02 − 1.77i)4-s + (0.762 + 0.440i)5-s − 1.53i·6-s − 1.84·8-s + (−0.115 + 0.200i)9-s + (1.33 − 0.769i)10-s + (−0.184 − 0.320i)11-s + (−1.55 − 0.900i)12-s + 1.84i·13-s + 0.771·15-s + (−0.583 + 1.01i)16-s + (0.214 − 0.123i)17-s + (0.202 + 0.350i)18-s + (−0.669 − 0.386i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.580 + 0.814i$
Analytic conductor: \(5.06512\)
Root analytic conductor: \(2.25058\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :2),\ -0.580 + 0.814i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.33090 - 2.58356i\)
\(L(\frac12)\) \(\approx\) \(1.33090 - 2.58356i\)
\(L(3)\) 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 \( 1 + (-3.49 + 6.05i)T + (-8 - 13.8i)T^{2} \)
3 \( 1 + (-6.83 + 3.94i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-19.0 - 11.0i)T + (312.5 + 541. i)T^{2} \)
11 \( 1 + (22.3 + 38.7i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 - 311. iT - 2.85e4T^{2} \)
17 \( 1 + (-61.9 + 35.7i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (241. + 139. i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-257. + 445. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 - 866.T + 7.07e5T^{2} \)
31 \( 1 + (-591. + 341. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (972. - 1.68e3i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + 102. iT - 2.82e6T^{2} \)
43 \( 1 + 885.T + 3.41e6T^{2} \)
47 \( 1 + (2.44e3 + 1.41e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-664. - 1.15e3i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (4.19e3 - 2.42e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-340. - 196. i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (2.12e3 + 3.67e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 1.18e3T + 2.54e7T^{2} \)
73 \( 1 + (-0.185 + 0.107i)T + (1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-789. + 1.36e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 9.05e3iT - 4.74e7T^{2} \)
89 \( 1 + (-3.01e3 - 1.74e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 7.94e3iT - 8.85e7T^{2} \)
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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

−13.81819280247990505216653112386, −13.60293893030318109373264181914, −12.19989185184718686245873483223, −11.08036195024907638148311285757, −9.980450779752621777951906582894, −8.708060883842448449125389574889, −6.51954589386798091152710183799, −4.66248980551911172117393981239, −2.86746406499145762531711899239, −1.84671143994611296741482819119, 3.34575391305508941448692900192, 5.03150832092460139812979171112, 6.09902743229341897928070352935, 7.78003705800563249334500664784, 8.762687950880567947330912358280, 10.11205469559263120264408689839, 12.53673629032285658032586424896, 13.33751608073252881563732995701, 14.36368432173657463773080888238, 15.18852344835736804717374997254

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