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Properties

Label 1-28e2-784.531-r0-0-0
Degree $1$
Conductor $784$
Sign $0.151 + 0.988i$
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  + (0.781 + 0.623i)3-s + (0.781 + 0.623i)5-s + (0.222 + 0.974i)9-s + (−0.974 − 0.222i)11-s + (0.974 + 0.222i)13-s + (0.222 + 0.974i)15-s + (0.900 − 0.433i)17-s − i·19-s + (−0.900 − 0.433i)23-s + (0.222 + 0.974i)25-s + (−0.433 + 0.900i)27-s + (−0.433 − 0.900i)29-s + 31-s + (−0.623 − 0.781i)33-s + (0.433 + 0.900i)37-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)3-s + (0.781 + 0.623i)5-s + (0.222 + 0.974i)9-s + (−0.974 − 0.222i)11-s + (0.974 + 0.222i)13-s + (0.222 + 0.974i)15-s + (0.900 − 0.433i)17-s − i·19-s + (−0.900 − 0.433i)23-s + (0.222 + 0.974i)25-s + (−0.433 + 0.900i)27-s + (−0.433 − 0.900i)29-s + 31-s + (−0.623 − 0.781i)33-s + (0.433 + 0.900i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.151 + 0.988i$
Analytic conductor: \(3.64088\)
Root analytic conductor: \(3.64088\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 784,\ (0:\ ),\ 0.151 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.660080686 + 1.424759834i\)
\(L(\frac12)\) \(\approx\) \(1.660080686 + 1.424759834i\)
\(L(1)\) \(\approx\) \(1.431036227 + 0.5885677330i\)
\(L(1)\) \(\approx\) \(1.431036227 + 0.5885677330i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.781 + 0.623i)T \)
5 \( 1 + (0.781 + 0.623i)T \)
11 \( 1 + (-0.974 - 0.222i)T \)
13 \( 1 + (0.974 + 0.222i)T \)
17 \( 1 + (0.900 - 0.433i)T \)
19 \( 1 - iT \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (-0.433 - 0.900i)T \)
31 \( 1 + T \)
37 \( 1 + (0.433 + 0.900i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (-0.781 + 0.623i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (-0.433 + 0.900i)T \)
59 \( 1 + (0.781 - 0.623i)T \)
61 \( 1 + (0.433 + 0.900i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 - T \)
83 \( 1 + (0.974 - 0.222i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
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

−21.86941625044058238606668512201, −21.16429523601534711010979248384, −20.54108288038613820248777037507, −19.86677993926847102394293341641, −18.894770856310702890697625414617, −18.01946170661019633504192424424, −17.66343850017330217766218268896, −16.405432519698251122771568804764, −15.65421845570677069846805538940, −14.6790995364653329438758682842, −13.80651577958855860942680072790, −13.13006297522506952413312286345, −12.6906348134631595872605395890, −11.570885744748118686782740610412, −10.30986776610487522732552432613, −9.6253207021285627903295198070, −8.6064108769485367636764866301, −8.09479840628033606390862078884, −7.04077507236575273693737507947, −6.00692929735708259976278616766, −5.22914835852092884712483247657, −3.934418041282938168819448117646, −2.85816089045949510841569045158, −1.930417880260255229721027734089, −0.96569048534563124220455602707, 1.58043699028414195105290826693, 2.62911561673436291063275844755, 3.33847854773054341047056089960, 4.40644966876100961289833443920, 5.57918753147861361992974416342, 6.27074314794468694206345124221, 7.68266607431782726800860185962, 8.21656429058910375707805852093, 9.36375193467114710992634711842, 10.10923229351777345591609373109, 10.60106168310966417222271454894, 11.672430312230760087160767474698, 13.00354643022709461993293600214, 13.74406230732997417922211268376, 14.25976077652740635679459185388, 15.13873587613913913359085787613, 16.00794850499667304167639408121, 16.62882264391488881819246198927, 17.82180942115324796456323653640, 18.71151129061729703463804237966, 19.02373576226449744848694480848, 20.49909478500276651817100822512, 20.88414423266384396475173368895, 21.46293391696536347455028687860, 22.44418078723784006736837864057

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