Properties

Label 2-7e2-49.11-c5-0-18
Degree $2$
Conductor $49$
Sign $0.119 + 0.992i$
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  + (7.04 − 6.54i)2-s + (25.9 − 3.90i)3-s + (4.51 − 60.3i)4-s + (−32.3 + 82.5i)5-s + (157. − 196. i)6-s + (−19.4 − 128. i)7-s + (−170. − 214. i)8-s + (423. − 130. i)9-s + (311. + 793. i)10-s + (−348. − 107. i)11-s + (−118. − 1.57e3i)12-s + (147. + 645. i)13-s + (−975. − 776. i)14-s + (−516. + 2.26e3i)15-s + (−689. − 103. i)16-s + (−445. + 303. i)17-s + ⋯
L(s)  = 1  + (1.24 − 1.15i)2-s + (1.66 − 0.250i)3-s + (0.141 − 1.88i)4-s + (−0.579 + 1.47i)5-s + (1.78 − 2.23i)6-s + (−0.150 − 0.988i)7-s + (−0.942 − 1.18i)8-s + (1.74 − 0.537i)9-s + (0.984 + 2.50i)10-s + (−0.868 − 0.267i)11-s + (−0.237 − 3.16i)12-s + (0.241 + 1.05i)13-s + (−1.33 − 1.05i)14-s + (−0.593 + 2.59i)15-s + (−0.673 − 0.101i)16-s + (−0.373 + 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.119 + 0.992i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.119 + 0.992i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ 0.119 + 0.992i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.30157 - 2.92820i\)
\(L(\frac12)\) \(\approx\) \(3.30157 - 2.92820i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (19.4 + 128. i)T \)
good2 \( 1 + (-7.04 + 6.54i)T + (2.39 - 31.9i)T^{2} \)
3 \( 1 + (-25.9 + 3.90i)T + (232. - 71.6i)T^{2} \)
5 \( 1 + (32.3 - 82.5i)T + (-2.29e3 - 2.12e3i)T^{2} \)
11 \( 1 + (348. + 107. i)T + (1.33e5 + 9.07e4i)T^{2} \)
13 \( 1 + (-147. - 645. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (445. - 303. i)T + (5.18e5 - 1.32e6i)T^{2} \)
19 \( 1 + (-903. - 1.56e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (2.98e3 + 2.03e3i)T + (2.35e6 + 5.99e6i)T^{2} \)
29 \( 1 + (-3.21e3 - 1.54e3i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (921. - 1.59e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-351. - 4.69e3i)T + (-6.85e7 + 1.03e7i)T^{2} \)
41 \( 1 + (7.72e3 + 9.68e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (-4.44e3 + 5.57e3i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (-1.21e4 + 1.12e4i)T + (1.71e7 - 2.28e8i)T^{2} \)
53 \( 1 + (-647. + 8.63e3i)T + (-4.13e8 - 6.23e7i)T^{2} \)
59 \( 1 + (1.38e4 + 3.52e4i)T + (-5.24e8 + 4.86e8i)T^{2} \)
61 \( 1 + (-541. - 7.22e3i)T + (-8.35e8 + 1.25e8i)T^{2} \)
67 \( 1 + (-954. + 1.65e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-1.87e4 + 9.05e3i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (-383. - 356. i)T + (1.54e8 + 2.06e9i)T^{2} \)
79 \( 1 + (-4.45e4 - 7.72e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-2.12e4 + 9.29e4i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (6.77e3 - 2.08e3i)T + (4.61e9 - 3.14e9i)T^{2} \)
97 \( 1 + 6.68e4T + 8.58e9T^{2} \)
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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

−14.01676607838094087767344755828, −13.63261648070155425186695553161, −12.20510148613439136635841654689, −10.80627175090772470091105910403, −10.06557735028044993619932614316, −8.034079667472041032156104082246, −6.75856816401220496534233590146, −4.01419828130305687562519892430, −3.30293290194564860312076690321, −2.10623346026727335664172579812, 2.89948938811097464546633139794, 4.36366197383870018742351874473, 5.46428336306114034216529743996, 7.71644728059380522274730806557, 8.299297472300817237157551913540, 9.405676311010594126758484783518, 12.20467168308135556619311457968, 13.07826508548428197706653223594, 13.70057727006054265622218355148, 15.16942186127723435406757087696

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