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Properties

Label 1-967-967.510-r0-0-0
Degree $1$
Conductor $967$
Sign $0.112 + 0.993i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (0.00975 − 0.999i)2-s + (−0.477 + 0.878i)3-s + (−0.999 − 0.0195i)4-s + (0.592 + 0.805i)5-s + (0.874 + 0.485i)6-s + (0.996 − 0.0779i)7-s + (−0.0292 + 0.999i)8-s + (−0.544 − 0.838i)9-s + (0.811 − 0.584i)10-s + (−0.696 + 0.717i)11-s + (0.494 − 0.869i)12-s + (−0.696 − 0.717i)13-s + (−0.0682 − 0.997i)14-s + (−0.990 + 0.136i)15-s + (0.999 + 0.0390i)16-s + (0.0876 + 0.996i)17-s + ⋯
L(s)  = 1  + (0.00975 − 0.999i)2-s + (−0.477 + 0.878i)3-s + (−0.999 − 0.0195i)4-s + (0.592 + 0.805i)5-s + (0.874 + 0.485i)6-s + (0.996 − 0.0779i)7-s + (−0.0292 + 0.999i)8-s + (−0.544 − 0.838i)9-s + (0.811 − 0.584i)10-s + (−0.696 + 0.717i)11-s + (0.494 − 0.869i)12-s + (−0.696 − 0.717i)13-s + (−0.0682 − 0.997i)14-s + (−0.990 + 0.136i)15-s + (0.999 + 0.0390i)16-s + (0.0876 + 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.112 + 0.993i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (510, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.112 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7580664005 + 0.6773236652i\)
\(L(\frac12)\) \(\approx\) \(0.7580664005 + 0.6773236652i\)
\(L(1)\) \(\approx\) \(0.8903847346 + 0.08867478173i\)
\(L(1)\) \(\approx\) \(0.8903847346 + 0.08867478173i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.00975 - 0.999i)T \)
3 \( 1 + (-0.477 + 0.878i)T \)
5 \( 1 + (0.592 + 0.805i)T \)
7 \( 1 + (0.996 - 0.0779i)T \)
11 \( 1 + (-0.696 + 0.717i)T \)
13 \( 1 + (-0.696 - 0.717i)T \)
17 \( 1 + (0.0876 + 0.996i)T \)
19 \( 1 + (0.389 + 0.921i)T \)
23 \( 1 + (0.909 - 0.416i)T \)
29 \( 1 + (0.316 + 0.948i)T \)
31 \( 1 + (-0.967 - 0.250i)T \)
37 \( 1 + (0.126 - 0.991i)T \)
41 \( 1 + (-0.576 - 0.816i)T \)
43 \( 1 + (0.353 + 0.935i)T \)
47 \( 1 + (-0.511 + 0.859i)T \)
53 \( 1 + (0.203 + 0.979i)T \)
59 \( 1 + (0.892 - 0.451i)T \)
61 \( 1 + (-0.576 - 0.816i)T \)
67 \( 1 + (-0.967 + 0.250i)T \)
71 \( 1 + (0.00975 + 0.999i)T \)
73 \( 1 + (0.203 - 0.979i)T \)
79 \( 1 + (-0.608 + 0.793i)T \)
83 \( 1 + (-0.668 + 0.744i)T \)
89 \( 1 + (-0.967 + 0.250i)T \)
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

−21.65726864178212028047175933556, −21.03715886095427282214653766710, −19.842332195391863196855719936888, −18.85550339012309289255268799777, −18.15225027316570749821007558096, −17.59903935249034755902177956023, −16.833700920118984852613337572940, −16.35664457497102412157225938996, −15.277558421998083485956445816236, −14.234716419024621470469690485476, −13.5416843827135061891670203984, −13.17758589592697644558035990363, −11.99465241258477862345096355578, −11.375121619839504809860914394065, −10.07579971199036444318047333188, −9.0160603020243796539223460639, −8.40542156207836111241931787566, −7.49886645805593678507840753151, −6.86193507792695547002402172588, −5.70506790105352551973225732093, −5.13089536028586534434897857232, −4.63160390011203504133953077646, −2.75478732970506200977139635437, −1.54163981962719549969347157531, −0.49278063198546690065683839688, 1.40056788521646553994434741351, 2.4429918532207248064140544231, 3.333216921215117379783573395440, 4.34526411123302297608572650929, 5.23770712956538839911080570891, 5.738469044093220948772522780212, 7.27667187504377550045656731546, 8.25441875795892355378148964106, 9.350340206862753851025060056, 10.11195355806498385275569160034, 10.70759001179231750704517981158, 11.08922938437826468883286036109, 12.29441040704856654231433519644, 12.84300273148430105532169215707, 14.22767316299912993870447968097, 14.65713399845158810631871511063, 15.272594260312561226337639311022, 16.721325734407225876621882098899, 17.51162391794462214619003331085, 17.926493391813053882483318007538, 18.644787662643654290757646259985, 19.8203862446829938599317621743, 20.6501823934423472970528964727, 21.12524387081568964123588137517, 21.811936177684283233301907879039

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