Properties

Label 2-7e2-49.11-c5-0-11
Degree $2$
Conductor $49$
Sign $0.696 + 0.717i$
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  + (−6.71 + 6.23i)2-s + (−0.814 + 0.122i)3-s + (3.88 − 51.8i)4-s + (−38.8 + 98.9i)5-s + (4.70 − 5.89i)6-s + (−59.4 − 115. i)7-s + (114. + 143. i)8-s + (−231. + 71.4i)9-s + (−355. − 906. i)10-s + (214. + 66.2i)11-s + (3.19 + 42.6i)12-s + (5.08 + 22.2i)13-s + (1.11e3 + 403. i)14-s + (19.4 − 85.3i)15-s + (−13.6 − 2.05i)16-s + (1.50e3 − 1.02e3i)17-s + ⋯
L(s)  = 1  + (−1.18 + 1.10i)2-s + (−0.0522 + 0.00787i)3-s + (0.121 − 1.61i)4-s + (−0.694 + 1.76i)5-s + (0.0533 − 0.0668i)6-s + (−0.458 − 0.888i)7-s + (0.630 + 0.790i)8-s + (−0.952 + 0.293i)9-s + (−1.12 − 2.86i)10-s + (0.535 + 0.165i)11-s + (0.00640 + 0.0855i)12-s + (0.00834 + 0.0365i)13-s + (1.52 + 0.550i)14-s + (0.0223 − 0.0978i)15-s + (−0.0132 − 0.00200i)16-s + (1.26 − 0.861i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.696 + 0.717i$
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: \((2,\ 49,\ (\ :5/2),\ 0.696 + 0.717i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.129364 - 0.0546914i\)
\(L(\frac12)\) \(\approx\) \(0.129364 - 0.0546914i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (59.4 + 115. i)T \)
good2 \( 1 + (6.71 - 6.23i)T + (2.39 - 31.9i)T^{2} \)
3 \( 1 + (0.814 - 0.122i)T + (232. - 71.6i)T^{2} \)
5 \( 1 + (38.8 - 98.9i)T + (-2.29e3 - 2.12e3i)T^{2} \)
11 \( 1 + (-214. - 66.2i)T + (1.33e5 + 9.07e4i)T^{2} \)
13 \( 1 + (-5.08 - 22.2i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (-1.50e3 + 1.02e3i)T + (5.18e5 - 1.32e6i)T^{2} \)
19 \( 1 + (10.9 + 18.9i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-600. - 409. i)T + (2.35e6 + 5.99e6i)T^{2} \)
29 \( 1 + (5.06e3 + 2.43e3i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (-1.88e3 + 3.26e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-199. - 2.66e3i)T + (-6.85e7 + 1.03e7i)T^{2} \)
41 \( 1 + (8.93e3 + 1.11e4i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (-6.35e3 + 7.96e3i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (1.86e4 - 1.73e4i)T + (1.71e7 - 2.28e8i)T^{2} \)
53 \( 1 + (1.20e3 - 1.60e4i)T + (-4.13e8 - 6.23e7i)T^{2} \)
59 \( 1 + (5.77e3 + 1.47e4i)T + (-5.24e8 + 4.86e8i)T^{2} \)
61 \( 1 + (3.40e3 + 4.54e4i)T + (-8.35e8 + 1.25e8i)T^{2} \)
67 \( 1 + (2.62e4 - 4.53e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-1.97e4 + 9.50e3i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (3.86e3 + 3.58e3i)T + (1.54e8 + 2.06e9i)T^{2} \)
79 \( 1 + (4.82e4 + 8.35e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-1.41e4 + 6.19e4i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (4.26e4 - 1.31e4i)T + (4.61e9 - 3.14e9i)T^{2} \)
97 \( 1 + 6.21e4T + 8.58e9T^{2} \)
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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

−14.67247426423291847217322938838, −14.02270590304988288217839988208, −11.62223160809956245127012365163, −10.57962737145549193467063553615, −9.582899118687065552132233224936, −7.84031432914153230067417117781, −7.18206072777474394112851284757, −6.11448828528203076068209603884, −3.33456365556749082805291236089, −0.11813523480110541277983150610, 1.30651652983652317151994370536, 3.42792882900918017081531030339, 5.54091858024932919539054445275, 8.201616282139643414962159372244, 8.776337076137251920307519474165, 9.669759359713633161079921384650, 11.43361274359353590521899345482, 12.14196670434804208981138993387, 12.82347146467440577638971420246, 14.93548152092848326595753710138

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