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Properties

Label 2-7e2-7.6-c6-0-4
Degree $2$
Conductor $49$
Sign $-0.755 - 0.654i$
Analytic cond. $11.2726$
Root an. cond. $3.35747$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 12·2-s + 12.1i·3-s + 80·4-s + 181. i·5-s − 145. i·6-s − 192·8-s + 582·9-s − 2.18e3i·10-s + 1.47e3·11-s + 969. i·12-s + 484. i·13-s − 2.20e3·15-s − 2.81e3·16-s + 3.01e3i·17-s − 6.98e3·18-s + 6.87e3i·19-s + ⋯
L(s)  = 1  − 1.5·2-s + 0.449i·3-s + 1.25·4-s + 1.45i·5-s − 0.673i·6-s − 0.375·8-s + 0.798·9-s − 2.18i·10-s + 1.11·11-s + 0.561i·12-s + 0.220i·13-s − 0.653·15-s − 0.687·16-s + 0.614i·17-s − 1.19·18-s + 1.00i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.755 - 0.654i$
Analytic conductor: \(11.2726\)
Root analytic conductor: \(3.35747\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3),\ -0.755 - 0.654i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.266735 + 0.715443i\)
\(L(\frac12)\) \(\approx\) \(0.266735 + 0.715443i\)
\(L(4)\) 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 \( 1 + 12T + 64T^{2} \)
3 \( 1 - 12.1iT - 729T^{2} \)
5 \( 1 - 181. iT - 1.56e4T^{2} \)
11 \( 1 - 1.47e3T + 1.77e6T^{2} \)
13 \( 1 - 484. iT - 4.82e6T^{2} \)
17 \( 1 - 3.01e3iT - 2.41e7T^{2} \)
19 \( 1 - 6.87e3iT - 4.70e7T^{2} \)
23 \( 1 + 5.91e3T + 1.48e8T^{2} \)
29 \( 1 - 3.97e3T + 5.94e8T^{2} \)
31 \( 1 - 1.28e4iT - 8.87e8T^{2} \)
37 \( 1 + 6.15e4T + 2.56e9T^{2} \)
41 \( 1 + 1.10e5iT - 4.75e9T^{2} \)
43 \( 1 + 1.74e4T + 6.32e9T^{2} \)
47 \( 1 + 3.06e4iT - 1.07e10T^{2} \)
53 \( 1 + 6.05e4T + 2.21e10T^{2} \)
59 \( 1 - 2.15e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.62e5iT - 5.15e10T^{2} \)
67 \( 1 + 2.68e5T + 9.04e10T^{2} \)
71 \( 1 - 1.01e5T + 1.28e11T^{2} \)
73 \( 1 + 3.17e5iT - 1.51e11T^{2} \)
79 \( 1 - 3.62e5T + 2.43e11T^{2} \)
83 \( 1 - 2.16e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.33e6iT - 4.96e11T^{2} \)
97 \( 1 - 1.51e6iT - 8.32e11T^{2} \)
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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

−15.04010621949871852735246434324, −14.00513791113223804225368107722, −11.96354124776651058735061953977, −10.62326965612425487298286265025, −10.15672681276811133704987700446, −8.940711179047486676003580615923, −7.44319370427491552486131129137, −6.50466424085863227894315048370, −3.80103635563573510036716314498, −1.71501057957441228679450528033, 0.61285274376595131798757964622, 1.59674486255865772407358992621, 4.62122566597817528276383066701, 6.74761412627232764767027227554, 8.003749362503105504736183812476, 9.047036786990555317737329427627, 9.809314670971383227725716319706, 11.45994244414498558376313660711, 12.57223016689697799374055282623, 13.67938125732688828552228251766

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