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Properties

Label 1-1617-1617.677-r1-0-0
Degree $1$
Conductor $1617$
Sign $0.0954 + 0.995i$
Analytic cond. $173.770$
Root an. cond. $173.770$
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  + (−0.646 − 0.762i)2-s + (−0.163 + 0.986i)4-s + (0.992 − 0.119i)5-s + (0.858 − 0.512i)8-s + (−0.733 − 0.680i)10-s + (0.983 − 0.178i)13-s + (−0.946 − 0.323i)16-s + (−0.251 + 0.967i)17-s + (0.669 − 0.743i)19-s + (−0.0448 + 0.998i)20-s + (−0.365 + 0.930i)23-s + (0.971 − 0.237i)25-s + (−0.772 − 0.635i)26-s + (−0.550 − 0.834i)29-s + (0.104 + 0.994i)31-s + (0.365 + 0.930i)32-s + ⋯
L(s)  = 1  + (−0.646 − 0.762i)2-s + (−0.163 + 0.986i)4-s + (0.992 − 0.119i)5-s + (0.858 − 0.512i)8-s + (−0.733 − 0.680i)10-s + (0.983 − 0.178i)13-s + (−0.946 − 0.323i)16-s + (−0.251 + 0.967i)17-s + (0.669 − 0.743i)19-s + (−0.0448 + 0.998i)20-s + (−0.365 + 0.930i)23-s + (0.971 − 0.237i)25-s + (−0.772 − 0.635i)26-s + (−0.550 − 0.834i)29-s + (0.104 + 0.994i)31-s + (0.365 + 0.930i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0954 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0954 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.0954 + 0.995i$
Analytic conductor: \(173.770\)
Root analytic conductor: \(173.770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1617,\ (1:\ ),\ 0.0954 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7635844734 + 0.6938953243i\)
\(L(\frac12)\) \(\approx\) \(0.7635844734 + 0.6938953243i\)
\(L(1)\) \(\approx\) \(0.8591117663 - 0.1501654727i\)
\(L(1)\) \(\approx\) \(0.8591117663 - 0.1501654727i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.646 - 0.762i)T \)
5 \( 1 + (0.992 - 0.119i)T \)
13 \( 1 + (0.983 - 0.178i)T \)
17 \( 1 + (-0.251 + 0.967i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.365 + 0.930i)T \)
29 \( 1 + (-0.550 - 0.834i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.447 + 0.894i)T \)
41 \( 1 + (-0.858 + 0.512i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.925 + 0.379i)T \)
53 \( 1 + (-0.712 + 0.701i)T \)
59 \( 1 + (-0.873 + 0.486i)T \)
61 \( 1 + (0.712 + 0.701i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.0448 + 0.998i)T \)
73 \( 1 + (-0.925 - 0.379i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (-0.983 - 0.178i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−20.25152403134340102140220770975, −18.944394343762097549619712157137, −18.39525859123448662878528636805, −17.97469678628698182980485722278, −17.10801697963121779258081093644, −16.32681825176295258952891681423, −15.91985570208595962819577649701, −14.784028726004760968236808414577, −14.17999513967254189079197595165, −13.60298288158438810363907057075, −12.7708433945057070315184711626, −11.49209217460304519516491747364, −10.766888569172393630970542737280, −9.96243172154195296302983339371, −9.33252417424853781421242679777, −8.641223908201447030616458195854, −7.74664892086825679234610924602, −6.826231011336424712240662595, −6.18164246361652047385063468696, −5.46914863339020115965947198923, −4.655482784492139149069677896685, −3.36578289489118683060897351514, −2.10942264217051574260629562632, −1.37770232600231661738736182034, −0.23189888411448065541484019168, 1.20381418670256133768636288193, 1.67522470206099464771770170052, 2.79390496794530897592393154199, 3.56678149965004817439162047968, 4.61019640281722823315135802652, 5.64828730199434876629109581605, 6.4981009932817913960030009157, 7.45899995947905525181850897956, 8.4837324600697467265062534412, 8.96971535676508931829848658222, 9.8875772846876963290524623666, 10.40699989536896102106369318120, 11.26291891131211638776164648158, 11.96234733095663652517539535996, 13.084015882524970132623411320700, 13.34811932972442246435788484646, 14.155987516080171699702919504553, 15.412618616816643231231891492830, 16.09274486734223045479441476385, 17.11431372850154585511055465011, 17.50990004189107925887635702692, 18.19624867494292242958341950077, 18.89886193020846104234338094926, 19.7466192486111582446973883510, 20.463502850767287637334526253691

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