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Properties

Label 2-28e2-7.3-c2-0-16
Degree $2$
Conductor $784$
Sign $-0.654 + 0.756i$
Analytic cond. $21.3624$
Root an. cond. $4.62195$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.86 − 2.23i)3-s + (−7.33 + 4.23i)5-s + (5.44 + 9.43i)9-s + (1.94 − 3.37i)11-s + 19.1i·13-s + 37.7·15-s + (11.5 + 6.69i)17-s + (−5.80 + 3.35i)19-s + (13 + 22.5i)23-s + (23.3 − 40.5i)25-s − 8.47i·27-s − 11.7·29-s + (−31.6 − 18.2i)31-s + (−15.0 + 8.69i)33-s + (16 + 27.7i)37-s + ⋯
L(s)  = 1  + (−1.28 − 0.743i)3-s + (−1.46 + 0.847i)5-s + (0.605 + 1.04i)9-s + (0.177 − 0.307i)11-s + 1.47i·13-s + 2.51·15-s + (0.681 + 0.393i)17-s + (−0.305 + 0.176i)19-s + (0.565 + 0.978i)23-s + (0.935 − 1.62i)25-s − 0.313i·27-s − 0.406·29-s + (−1.02 − 0.590i)31-s + (−0.456 + 0.263i)33-s + (0.432 + 0.748i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.756i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.654 + 0.756i$
Analytic conductor: \(21.3624\)
Root analytic conductor: \(4.62195\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1),\ -0.654 + 0.756i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1126630865\)
\(L(\frac12)\) \(\approx\) \(0.1126630865\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (3.86 + 2.23i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (7.33 - 4.23i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-1.94 + 3.37i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 19.1iT - 169T^{2} \)
17 \( 1 + (-11.5 - 6.69i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (5.80 - 3.35i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-13 - 22.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 11.7T + 841T^{2} \)
31 \( 1 + (31.6 + 18.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-16 - 27.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 20.9iT - 1.68e3T^{2} \)
43 \( 1 + 79.2T + 1.84e3T^{2} \)
47 \( 1 + (-12.3 + 7.15i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-6.89 + 11.9i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-7.31 - 4.22i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-27.4 + 15.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (15.6 - 27.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 95.5T + 5.04e3T^{2} \)
73 \( 1 + (15.8 + 9.15i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-39.8 - 69.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 141. iT - 6.88e3T^{2} \)
89 \( 1 + (100. - 58.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 137. iT - 9.40e3T^{2} \)
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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

−10.05846011555651775932460828670, −8.781033279920545045118041466754, −7.70665786172564858682440440976, −7.08236659389910431204149046152, −6.48194091140878423484490414288, −5.49082166773548514067887522460, −4.25810355306621333820852840230, −3.37042291022300689909091204268, −1.59526659156477290739723734856, −0.06823996331094978610625698543, 0.838075264076784926266540016851, 3.27619491436710267901266581058, 4.27333909027516529531784988605, 5.00539523303944945764172844127, 5.64938179907800395525042808619, 6.95372710020214499926670973663, 7.86768952214700950008597889259, 8.658270886512302628238599947536, 9.717515485228711826203281100804, 10.64159987601249164393552090530

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