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Properties

Label 2-28e2-16.5-c1-0-51
Degree $2$
Conductor $784$
Sign $0.882 - 0.470i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.19 + 0.756i)2-s + (1.42 − 1.42i)3-s + (0.854 + 1.80i)4-s + (0.702 + 0.702i)5-s + (2.78 − 0.625i)6-s + (−0.348 + 2.80i)8-s − 1.08i·9-s + (0.307 + 1.37i)10-s + (−1.38 − 1.38i)11-s + (3.80 + 1.36i)12-s + (2.10 − 2.10i)13-s + 2.00·15-s + (−2.54 + 3.08i)16-s + 5.67·17-s + (0.822 − 1.29i)18-s + (0.451 − 0.451i)19-s + ⋯
L(s)  = 1  + (0.844 + 0.535i)2-s + (0.825 − 0.825i)3-s + (0.427 + 0.904i)4-s + (0.313 + 0.313i)5-s + (1.13 − 0.255i)6-s + (−0.123 + 0.992i)8-s − 0.362i·9-s + (0.0971 + 0.433i)10-s + (−0.416 − 0.416i)11-s + (1.09 + 0.393i)12-s + (0.583 − 0.583i)13-s + 0.518·15-s + (−0.635 + 0.772i)16-s + 1.37·17-s + (0.193 − 0.306i)18-s + (0.103 − 0.103i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.882 - 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.32259 + 0.830406i\)
\(L(\frac12)\) \(\approx\) \(3.32259 + 0.830406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 - 0.756i)T \)
7 \( 1 \)
good3 \( 1 + (-1.42 + 1.42i)T - 3iT^{2} \)
5 \( 1 + (-0.702 - 0.702i)T + 5iT^{2} \)
11 \( 1 + (1.38 + 1.38i)T + 11iT^{2} \)
13 \( 1 + (-2.10 + 2.10i)T - 13iT^{2} \)
17 \( 1 - 5.67T + 17T^{2} \)
19 \( 1 + (-0.451 + 0.451i)T - 19iT^{2} \)
23 \( 1 - 6.84iT - 23T^{2} \)
29 \( 1 + (0.207 - 0.207i)T - 29iT^{2} \)
31 \( 1 + 7.89T + 31T^{2} \)
37 \( 1 + (7.03 + 7.03i)T + 37iT^{2} \)
41 \( 1 + 2.40iT - 41T^{2} \)
43 \( 1 + (-3.65 - 3.65i)T + 43iT^{2} \)
47 \( 1 + 0.289T + 47T^{2} \)
53 \( 1 + (5.58 + 5.58i)T + 53iT^{2} \)
59 \( 1 + (9.84 + 9.84i)T + 59iT^{2} \)
61 \( 1 + (-2.64 + 2.64i)T - 61iT^{2} \)
67 \( 1 + (7.35 - 7.35i)T - 67iT^{2} \)
71 \( 1 - 11.5iT - 71T^{2} \)
73 \( 1 - 0.358iT - 73T^{2} \)
79 \( 1 - 7.69T + 79T^{2} \)
83 \( 1 + (-0.424 + 0.424i)T - 83iT^{2} \)
89 \( 1 + 17.5iT - 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−10.50417429549229699405725373663, −9.244923773805274765071020327928, −8.245165035622159157588599640237, −7.70031465823484582712481321693, −7.01995386159276081206648150982, −5.86774900136331464097340875891, −5.29467200530568184274948350301, −3.62871707598285840631228155492, −2.98997025320225414827683745059, −1.77985338165153939733378736468, 1.55745683715994428157896525998, 2.88896645227206919731104808131, 3.70138448381045808032196846264, 4.61618056074856154302852545336, 5.47545246222642784615456355152, 6.51040804156941449459599816069, 7.70021229340976863571126524478, 8.906953578138277833941494643726, 9.456663126507585907707061169758, 10.29813874430930653152788221529

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