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Properties

Label 1-28e2-784.355-r0-0-0
Degree $1$
Conductor $784$
Sign $0.999 + 0.00801i$
Analytic cond. $3.64088$
Root an. cond. $3.64088$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.997 + 0.0747i)3-s + (−0.563 + 0.826i)5-s + (0.988 + 0.149i)9-s + (−0.149 − 0.988i)11-s + (0.781 − 0.623i)13-s + (−0.623 + 0.781i)15-s + (−0.955 − 0.294i)17-s + (0.866 − 0.5i)19-s + (0.955 − 0.294i)23-s + (−0.365 − 0.930i)25-s + (0.974 + 0.222i)27-s + (0.974 − 0.222i)29-s + (−0.5 + 0.866i)31-s + (−0.0747 − 0.997i)33-s + (0.680 + 0.733i)37-s + ⋯
L(s)  = 1  + (0.997 + 0.0747i)3-s + (−0.563 + 0.826i)5-s + (0.988 + 0.149i)9-s + (−0.149 − 0.988i)11-s + (0.781 − 0.623i)13-s + (−0.623 + 0.781i)15-s + (−0.955 − 0.294i)17-s + (0.866 − 0.5i)19-s + (0.955 − 0.294i)23-s + (−0.365 − 0.930i)25-s + (0.974 + 0.222i)27-s + (0.974 − 0.222i)29-s + (−0.5 + 0.866i)31-s + (−0.0747 − 0.997i)33-s + (0.680 + 0.733i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.999 + 0.00801i$
Analytic conductor: \(3.64088\)
Root analytic conductor: \(3.64088\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 784,\ (0:\ ),\ 0.999 + 0.00801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.999883121 + 0.008013841519i\)
\(L(\frac12)\) \(\approx\) \(1.999883121 + 0.008013841519i\)
\(L(1)\) \(\approx\) \(1.435675967 + 0.06052251388i\)
\(L(1)\) \(\approx\) \(1.435675967 + 0.06052251388i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.997 + 0.0747i)T \)
5 \( 1 + (-0.563 + 0.826i)T \)
11 \( 1 + (-0.149 - 0.988i)T \)
13 \( 1 + (0.781 - 0.623i)T \)
17 \( 1 + (-0.955 - 0.294i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (0.974 - 0.222i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.680 + 0.733i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (0.433 - 0.900i)T \)
47 \( 1 + (0.365 - 0.930i)T \)
53 \( 1 + (-0.680 + 0.733i)T \)
59 \( 1 + (-0.563 - 0.826i)T \)
61 \( 1 + (0.680 + 0.733i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.365 + 0.930i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.781 + 0.623i)T \)
89 \( 1 + (-0.988 - 0.149i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.30456525669599198274779991731, −21.1481806914982702243005710910, −20.66309173080035163215996949061, −19.96816322469962278984304093103, −19.32182860955700525144542296672, −18.423139833082109373105407102866, −17.584965648568363861927692239497, −16.443763210920441685132629552598, −15.73267595793127521094261311773, −15.11669882131897744319965725658, −14.160759192415340422999077716793, −13.24409755954959924347097021348, −12.694946684329103248771115352349, −11.762741273867847889403856016620, −10.743772966196934506885558679, −9.45779076205156148658685873057, −9.08635884117351501035025996570, −8.07945132724292526852664051514, −7.43845223317535555121422266407, −6.42751665093464065617418882026, −4.95689087562723167260815076786, −4.235572003566357158647383506583, −3.39582762574909514767199027201, −2.10942518914987146509871356289, −1.20720994865030490017863231713, 1.00695091794429991258476635533, 2.62721707056593323022442341319, 3.13324512780846378660607061603, 4.000422796001193917301688331353, 5.17642206680205402590156535163, 6.542451623844132077027494019134, 7.19658894616854927541931286488, 8.284924211818377013813854913987, 8.70908215890440434047861697320, 9.88982475381355037721985289465, 10.81982463208725965215027947000, 11.38238709012518699562257977346, 12.65489386460690813958711237216, 13.65296268826921429417782853148, 14.007008013587599478666898434138, 15.20227846473482037708652073397, 15.57432358792472762336151566137, 16.359286265603694322755442693633, 17.78590481142646127683320433214, 18.51542531760716672092999378540, 19.07698475020892917478878173954, 19.99942374226741659087451254079, 20.47954136971685631775119193790, 21.65528126770677281267779842677, 22.10039663844365904301012976755

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