Properties

Label 1-967-967.208-r1-0-0
Degree $1$
Conductor $967$
Sign $0.390 - 0.920i$
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  + (0.924 + 0.380i)2-s + (−0.494 − 0.869i)3-s + (0.710 + 0.703i)4-s + (−0.972 + 0.232i)5-s + (−0.126 − 0.991i)6-s + (0.999 + 0.0195i)7-s + (0.389 + 0.921i)8-s + (−0.511 + 0.859i)9-s + (−0.987 − 0.155i)10-s + (0.833 − 0.552i)11-s + (0.260 − 0.965i)12-s + (−0.833 − 0.552i)13-s + (0.917 + 0.398i)14-s + (0.682 + 0.730i)15-s + (0.00975 + 0.999i)16-s + (−0.932 + 0.362i)17-s + ⋯
L(s)  = 1  + (0.924 + 0.380i)2-s + (−0.494 − 0.869i)3-s + (0.710 + 0.703i)4-s + (−0.972 + 0.232i)5-s + (−0.126 − 0.991i)6-s + (0.999 + 0.0195i)7-s + (0.389 + 0.921i)8-s + (−0.511 + 0.859i)9-s + (−0.987 − 0.155i)10-s + (0.833 − 0.552i)11-s + (0.260 − 0.965i)12-s + (−0.833 − 0.552i)13-s + (0.917 + 0.398i)14-s + (0.682 + 0.730i)15-s + (0.00975 + 0.999i)16-s + (−0.932 + 0.362i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.390 - 0.920i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.390 - 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.007309393 - 1.328761756i\)
\(L(\frac12)\) \(\approx\) \(2.007309393 - 1.328761756i\)
\(L(1)\) \(\approx\) \(1.424088241 - 0.09133716511i\)
\(L(1)\) \(\approx\) \(1.424088241 - 0.09133716511i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.924 - 0.380i)T \)
3 \( 1 + (0.494 + 0.869i)T \)
5 \( 1 + (0.972 - 0.232i)T \)
7 \( 1 + (-0.999 - 0.0195i)T \)
11 \( 1 + (-0.833 + 0.552i)T \)
13 \( 1 + (0.833 + 0.552i)T \)
17 \( 1 + (0.932 - 0.362i)T \)
19 \( 1 + (-0.957 + 0.288i)T \)
23 \( 1 + (-0.107 + 0.994i)T \)
29 \( 1 + (0.951 - 0.307i)T \)
31 \( 1 + (0.750 + 0.660i)T \)
37 \( 1 + (0.353 - 0.935i)T \)
41 \( 1 + (0.854 + 0.519i)T \)
43 \( 1 + (-0.297 - 0.954i)T \)
47 \( 1 + (-0.864 + 0.502i)T \)
53 \( 1 + (0.334 + 0.942i)T \)
59 \( 1 + (-0.993 - 0.116i)T \)
61 \( 1 + (-0.854 - 0.519i)T \)
67 \( 1 + (-0.750 + 0.660i)T \)
71 \( 1 + (-0.924 + 0.380i)T \)
73 \( 1 + (0.334 - 0.942i)T \)
79 \( 1 + (0.527 + 0.849i)T \)
83 \( 1 + (0.544 + 0.838i)T \)
89 \( 1 + (-0.750 + 0.660i)T \)
97 \( 1 + (-0.623 - 0.781i)T \)
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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

−21.93200402979056413920254536725, −20.92593185858472346562701434938, −20.226184492017356309149344633589, −19.847868732327181654468449511618, −18.72441581158985997860486184828, −17.56772198608647813978699912151, −16.82673720050672112258840619256, −15.88378813832516509730299542665, −15.318896753808846236627751010277, −14.58810476914756970948952001856, −14.00917537210537787672068864779, −12.6060628660870191731928584616, −11.81572308360726109543966101116, −11.500444179463772868152538317190, −10.79511366769038766399708560938, −9.63589171136822432602472407316, −8.96595037357320691466945882195, −7.47270401094970606827135904310, −6.88911518785542745324824219294, −5.44959717488937220376637675253, −4.94137801282057129105184880858, −4.10656189893615447703661600976, −3.61984475580622265935821748314, −2.16697981180466985318563519901, −1.0064880850915605098934597429, 0.43374265218254674253160579495, 1.77624446109754741870743510974, 2.80146000555574186291675556442, 3.91530042929647821467194180173, 4.84320524629608747109649381619, 5.56274681158066492257593591804, 6.69413764222587597588430792618, 7.2432607879578306408310303169, 8.03494290029437691458851842402, 8.69780507877322394181336159957, 10.641413969625072412189695982697, 11.41868713615554329076333208768, 11.72157655638726982995795093846, 12.59712103972302626173941479216, 13.381743941711296027014158956408, 14.40314411651530751390101500058, 14.80278702533063565958588785270, 15.7737828769686137691558189653, 16.73588527805672648886963693715, 17.290366422792616862327819498686, 18.18270569456657558452664528821, 19.07607312530156795694244231802, 20.06181445479635303737565276362, 20.398764706097652985514133981385, 21.97404024028723716009027517746

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