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Properties

Label 1-967-967.59-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.688 - 0.725i$
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  + (−0.347 + 0.937i)2-s + (−0.696 − 0.717i)3-s + (−0.759 − 0.651i)4-s + (−0.00325 − 0.999i)5-s + (0.914 − 0.404i)6-s + (0.216 + 0.976i)7-s + (0.874 − 0.485i)8-s + (−0.0292 + 0.999i)9-s + (0.938 + 0.344i)10-s + (−0.967 − 0.250i)11-s + (0.0617 + 0.998i)12-s + (0.266 − 0.963i)13-s + (−0.990 − 0.136i)14-s + (−0.715 + 0.699i)15-s + (0.152 + 0.988i)16-s + (0.0487 + 0.998i)17-s + ⋯
L(s)  = 1  + (−0.347 + 0.937i)2-s + (−0.696 − 0.717i)3-s + (−0.759 − 0.651i)4-s + (−0.00325 − 0.999i)5-s + (0.914 − 0.404i)6-s + (0.216 + 0.976i)7-s + (0.874 − 0.485i)8-s + (−0.0292 + 0.999i)9-s + (0.938 + 0.344i)10-s + (−0.967 − 0.250i)11-s + (0.0617 + 0.998i)12-s + (0.266 − 0.963i)13-s + (−0.990 − 0.136i)14-s + (−0.715 + 0.699i)15-s + (0.152 + 0.988i)16-s + (0.0487 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.688 - 0.725i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.688 - 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1211689578 - 0.2822249664i\)
\(L(\frac12)\) \(\approx\) \(0.1211689578 - 0.2822249664i\)
\(L(1)\) \(\approx\) \(0.5584632712 + 0.007898117012i\)
\(L(1)\) \(\approx\) \(0.5584632712 + 0.007898117012i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.347 + 0.937i)T \)
3 \( 1 + (-0.696 - 0.717i)T \)
5 \( 1 + (-0.00325 - 0.999i)T \)
7 \( 1 + (0.216 + 0.976i)T \)
11 \( 1 + (-0.967 - 0.250i)T \)
13 \( 1 + (0.266 - 0.963i)T \)
17 \( 1 + (0.0487 + 0.998i)T \)
19 \( 1 + (-0.922 + 0.386i)T \)
23 \( 1 + (0.592 - 0.805i)T \)
29 \( 1 + (0.763 - 0.646i)T \)
31 \( 1 + (0.978 - 0.206i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.334 - 0.942i)T \)
43 \( 1 + (0.999 - 0.0260i)T \)
47 \( 1 + (-0.992 - 0.123i)T \)
53 \( 1 + (0.113 + 0.993i)T \)
59 \( 1 + (-0.705 - 0.708i)T \)
61 \( 1 + (0.983 + 0.181i)T \)
67 \( 1 + (-0.668 + 0.744i)T \)
71 \( 1 + (-0.638 + 0.769i)T \)
73 \( 1 + (0.113 - 0.993i)T \)
79 \( 1 + (0.328 + 0.944i)T \)
83 \( 1 + (-0.597 - 0.801i)T \)
89 \( 1 + (-0.310 - 0.950i)T \)
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

−21.845261367758467071688895584135, −21.145937846727005447726902577960, −20.782030358001800712674348661, −19.65842300160036319260003527520, −18.92267492944130544134008493559, −17.97064797836410993047427595201, −17.61140422226581482930716511838, −16.66980132282162992811452634586, −15.89902207094796896315161919094, −14.87401932912166440982826752638, −13.92639136386414473403103641864, −13.28808500146770126482652806512, −12.06411654375016230066295798296, −11.28045037112823831536503674149, −10.83214028720425545400719131764, −10.11967961497577739754630638708, −9.515537647415752818691786797982, −8.33723004068660394339128215940, −7.22301546682498406470665249012, −6.563263285917193141620770190643, −5.02635245158089372110030431192, −4.45712336299373671483163258919, −3.45537138393716961859957837262, −2.66828873045865580943550142832, −1.2885190485374233421556227263, 0.18630002160550608147370289595, 1.323805350991055470139076222049, 2.4439914292384067351515598306, 4.32086897195225924689750703534, 5.21934807360629884385124663712, 5.763074758065492143139305457114, 6.39577490681385043402925910103, 7.748420897060364445191765002056, 8.30426072438918351438178691757, 8.770440264760094108212885345186, 10.19345158864466702991508091759, 10.79620947163544199856610277728, 12.123839023554933446639861331, 12.8190925882605938610341721181, 13.23635028004987867269002799096, 14.39820956802987868929211217002, 15.53312369276960930396634664544, 15.84611413431044818321457565208, 16.9194995697919967565672422213, 17.43261959199891957853231566321, 18.10646817188504493218819228951, 18.989628328396845232536636111353, 19.39940988734911871136974103613, 20.78851179982939577823681536033, 21.51054940511855415985638060932

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