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Properties

Label 1-967-967.102-r1-0-0
Degree $1$
Conductor $967$
Sign $0.151 - 0.988i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (−0.658 + 0.752i)2-s + (0.864 + 0.502i)3-s + (−0.132 − 0.991i)4-s + (0.597 + 0.801i)5-s + (−0.947 + 0.319i)6-s + (0.870 − 0.491i)7-s + (0.833 + 0.552i)8-s + (0.494 + 0.869i)9-s + (−0.996 − 0.0779i)10-s + (−0.957 + 0.288i)11-s + (0.383 − 0.923i)12-s + (−0.228 − 0.973i)13-s + (−0.203 + 0.979i)14-s + (0.113 + 0.993i)15-s + (−0.964 + 0.263i)16-s + (−0.184 − 0.982i)17-s + ⋯
L(s)  = 1  + (−0.658 + 0.752i)2-s + (0.864 + 0.502i)3-s + (−0.132 − 0.991i)4-s + (0.597 + 0.801i)5-s + (−0.947 + 0.319i)6-s + (0.870 − 0.491i)7-s + (0.833 + 0.552i)8-s + (0.494 + 0.869i)9-s + (−0.996 − 0.0779i)10-s + (−0.957 + 0.288i)11-s + (0.383 − 0.923i)12-s + (−0.228 − 0.973i)13-s + (−0.203 + 0.979i)14-s + (0.113 + 0.993i)15-s + (−0.964 + 0.263i)16-s + (−0.184 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.151 - 0.988i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.151 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3121141961 - 0.2679756126i\)
\(L(\frac12)\) \(\approx\) \(0.3121141961 - 0.2679756126i\)
\(L(1)\) \(\approx\) \(0.8509290616 + 0.4154910516i\)
\(L(1)\) \(\approx\) \(0.8509290616 + 0.4154910516i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.658 + 0.752i)T \)
3 \( 1 + (0.864 + 0.502i)T \)
5 \( 1 + (0.597 + 0.801i)T \)
7 \( 1 + (0.870 - 0.491i)T \)
11 \( 1 + (-0.957 + 0.288i)T \)
13 \( 1 + (-0.228 - 0.973i)T \)
17 \( 1 + (-0.184 - 0.982i)T \)
19 \( 1 + (-0.929 - 0.368i)T \)
23 \( 1 + (0.668 + 0.744i)T \)
29 \( 1 + (-0.987 + 0.155i)T \)
31 \( 1 + (-0.986 + 0.161i)T \)
37 \( 1 + (-0.903 - 0.428i)T \)
41 \( 1 + (-0.962 - 0.269i)T \)
43 \( 1 + (-0.401 - 0.915i)T \)
47 \( 1 + (0.705 - 0.708i)T \)
53 \( 1 + (-0.419 - 0.907i)T \)
59 \( 1 + (0.448 + 0.893i)T \)
61 \( 1 + (-0.247 - 0.968i)T \)
67 \( 1 + (-0.353 - 0.935i)T \)
71 \( 1 + (0.981 - 0.193i)T \)
73 \( 1 + (-0.419 + 0.907i)T \)
79 \( 1 + (0.0162 + 0.999i)T \)
83 \( 1 + (-0.522 - 0.852i)T \)
89 \( 1 + (-0.633 + 0.773i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−21.35796176356941536429805255835, −20.815112946223944840768690171032, −20.3492519671103267885714821293, −19.196413424579148841646708242306, −18.75807237144416418090518332726, −18.02105220282325082309201980393, −17.17642196564585863131307852400, −16.53413648299939814291070667789, −15.31740569208004285579676939919, −14.44530699991598475533186463963, −13.51051586921102799762408208591, −12.81644130977070378292314645884, −12.315962096020139057758091264852, −11.23371988156014897328144011672, −10.34286351514358137826446320331, −9.35907467868152738382492305659, −8.62870714931459574365648863040, −8.29962514376343747373783994591, −7.32380434366152896740598234602, −6.12570857446995362250221972732, −4.83539189767966716023135767902, −3.992322237980123913476935130996, −2.68143283704530065224517898038, −1.90769146779063698863897434850, −1.421706604055949979922618107372, 0.082923276752761965953329183896, 1.74069831772036256184509172198, 2.4533992738735582879665400436, 3.66135913560767386650711405936, 5.12013424202548899967502950186, 5.32464294520354241936391101192, 7.06330527183896183334428063863, 7.35585180466254866626021809782, 8.27824226236554804908220339844, 9.124742721253813895960912216747, 10.004557185328064056048558304083, 10.624886753336561201976964092636, 11.12427590795350068160965090025, 13.070574420228705172614869204736, 13.70079472153418300500829822837, 14.42482304592417559861817162969, 15.24352033849740860076218349919, 15.44592527279221156990419445576, 16.75627848606276730890473034762, 17.45834319983281629388806397698, 18.23477299193738813147845392016, 18.77349056402688149933804782392, 19.83837090526893820948400195317, 20.50260984468314089144536627838, 21.192782591791643686890610634309

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