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Properties

Label 1-693-693.292-r0-0-0
Degree $1$
Conductor $693$
Sign $-0.0349 - 0.999i$
Analytic cond. $3.21827$
Root an. cond. $3.21827$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.913 + 0.406i)5-s + (−0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (0.913 + 0.406i)13-s + (−0.809 − 0.587i)16-s + (0.913 − 0.406i)17-s + (0.669 − 0.743i)19-s + (0.104 + 0.994i)20-s + (−0.5 + 0.866i)23-s + (0.669 − 0.743i)25-s + (0.978 − 0.207i)26-s + (−0.669 − 0.743i)29-s + (0.809 − 0.587i)31-s − 32-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.913 + 0.406i)5-s + (−0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (0.913 + 0.406i)13-s + (−0.809 − 0.587i)16-s + (0.913 − 0.406i)17-s + (0.669 − 0.743i)19-s + (0.104 + 0.994i)20-s + (−0.5 + 0.866i)23-s + (0.669 − 0.743i)25-s + (0.978 − 0.207i)26-s + (−0.669 − 0.743i)29-s + (0.809 − 0.587i)31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0349 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0349 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.0349 - 0.999i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (292, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ -0.0349 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.333620041 - 1.381066091i\)
\(L(\frac12)\) \(\approx\) \(1.333620041 - 1.381066091i\)
\(L(1)\) \(\approx\) \(1.321288892 - 0.6391446122i\)
\(L(1)\) \(\approx\) \(1.321288892 - 0.6391446122i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−22.96997197282954982567720708044, −22.4036091706829497600788381037, −21.172726264278814484604062061931, −20.660064400817634591328504727730, −19.8584177493415661021103724080, −18.77522185874926662275587660751, −17.88774181083492237293387089418, −16.76798871667098134500575493713, −16.16821179979147612500264649001, −15.57743506641039397899542592308, −14.60210285508970578836202239337, −13.94836864973146023034898195993, −12.74322367034918505068898943504, −12.36530328075391900602673252205, −11.41813959233918041861454940364, −10.52017507874155970795046837866, −9.06027470257992365557931549474, −8.088358029280490225824415711238, −7.68698616245083974417023206271, −6.457860559316228953998076186567, −5.60956810529051368670959969458, −4.647375055025407916441793296021, −3.71869054352341909355100719035, −3.0516303986770163565192585489, −1.32535599225562719408507076527, 0.79895699051273178414319217974, 2.16386068309688659875438303791, 3.3606911689550873018614857500, 3.86118084205332461746554026868, 4.982860642014155784045117098588, 5.95433181701830354703179891950, 6.96788065978835494082254866171, 7.817308808472281516110596357884, 9.09892462811137710112537337741, 10.06109982767050835981025150258, 11.02751041752560795710321875059, 11.675228883231125008405674656967, 12.24019315027401702060191187934, 13.51794064452418114084658659792, 13.95732212506396022941860435278, 15.06946599760379605849537916057, 15.65700115203664013590274059994, 16.37606205775171942380372685574, 17.79089948978746330267207695306, 18.81768841243046484295411779532, 19.1730553904871353442986118750, 20.185010831443825928221342129792, 20.82360253611088573545840867603, 21.67666823823065024420769151267, 22.652388862179421669345577649384

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