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Properties

Label 1-1792-1792.619-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.624 - 0.780i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
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  + (0.162 − 0.986i)3-s + (0.973 + 0.227i)5-s + (−0.946 − 0.321i)9-s + (−0.910 + 0.412i)11-s + (0.956 + 0.290i)13-s + (0.382 − 0.923i)15-s + (0.608 − 0.793i)17-s + (0.999 + 0.0327i)19-s + (−0.442 − 0.896i)23-s + (0.896 + 0.442i)25-s + (−0.471 + 0.881i)27-s + (0.995 − 0.0980i)29-s + (−0.258 + 0.965i)31-s + (0.258 + 0.965i)33-s + (−0.849 + 0.528i)37-s + ⋯
L(s)  = 1  + (0.162 − 0.986i)3-s + (0.973 + 0.227i)5-s + (−0.946 − 0.321i)9-s + (−0.910 + 0.412i)11-s + (0.956 + 0.290i)13-s + (0.382 − 0.923i)15-s + (0.608 − 0.793i)17-s + (0.999 + 0.0327i)19-s + (−0.442 − 0.896i)23-s + (0.896 + 0.442i)25-s + (−0.471 + 0.881i)27-s + (0.995 − 0.0980i)29-s + (−0.258 + 0.965i)31-s + (0.258 + 0.965i)33-s + (−0.849 + 0.528i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.624 - 0.780i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.624 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.895189107 - 0.9106977595i\)
\(L(\frac12)\) \(\approx\) \(1.895189107 - 0.9106977595i\)
\(L(1)\) \(\approx\) \(1.304396324 - 0.3748001542i\)
\(L(1)\) \(\approx\) \(1.304396324 - 0.3748001542i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.162 - 0.986i)T \)
5 \( 1 + (0.973 + 0.227i)T \)
11 \( 1 + (-0.910 + 0.412i)T \)
13 \( 1 + (0.956 + 0.290i)T \)
17 \( 1 + (0.608 - 0.793i)T \)
19 \( 1 + (0.999 + 0.0327i)T \)
23 \( 1 + (-0.442 - 0.896i)T \)
29 \( 1 + (0.995 - 0.0980i)T \)
31 \( 1 + (-0.258 + 0.965i)T \)
37 \( 1 + (-0.849 + 0.528i)T \)
41 \( 1 + (0.831 + 0.555i)T \)
43 \( 1 + (0.773 - 0.634i)T \)
47 \( 1 + (-0.130 + 0.991i)T \)
53 \( 1 + (-0.412 - 0.910i)T \)
59 \( 1 + (0.729 + 0.683i)T \)
61 \( 1 + (-0.352 - 0.935i)T \)
67 \( 1 + (-0.986 - 0.162i)T \)
71 \( 1 + (0.195 - 0.980i)T \)
73 \( 1 + (0.659 + 0.751i)T \)
79 \( 1 + (-0.793 + 0.608i)T \)
83 \( 1 + (0.881 - 0.471i)T \)
89 \( 1 + (0.997 - 0.0654i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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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

−20.61177671978296418032019699806, −19.737407320977985111196206452473, −18.823339416986595873968127615850, −17.899280823339822671405594664791, −17.4407767836719277422441551084, −16.407748581269729197869599434896, −15.99656617220750429510466873414, −15.26393926351395488106398394294, −14.28417683987137731707634022882, −13.72101458752972654896972962180, −13.09572857471041336007661914771, −12.06188177453606618983577123460, −11.02497005681155259339087507886, −10.4540564073827763499429641627, −9.82610598321822208711228238098, −9.055047463614723564858189620800, −8.33160820014284181011444280025, −7.54146890225132006603029503254, −5.995284970588075176077237361082, −5.72728813664345359517091058258, −4.9450409662665304783647522709, −3.8331182233076826879158136416, −3.12105706213165177697406536937, −2.19183721390351893794719243837, −1.00144610455950022047027009709, 0.90571791604779315111005660371, 1.7581789178279542494589647254, 2.669504695160529232959897082239, 3.272186771122521528436991908247, 4.8117999840296500750825297501, 5.5813859904766978952017814733, 6.34000611501818284359704928234, 7.04500354013114757049987450638, 7.84962323772352995633783967368, 8.66443978467365144562415288398, 9.49306978569567755368735740620, 10.31965878209863934578000445778, 11.09553597863117324871379623722, 12.09763052415436862021427254637, 12.679886252887705185819797660602, 13.571610836357674368777881650073, 13.97656686188467021517647334574, 14.58355603237544036553027246103, 15.824226846688457769273398230008, 16.38501843053730661543335838794, 17.522224057070237524117952436103, 18.01042626093915522868692669427, 18.46126827927519378998342645336, 19.145327956485448126528815414914, 20.23728138319822965941295222797

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