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Properties

Label 1-967-967.180-r1-0-0
Degree $1$
Conductor $967$
Sign $0.315 + 0.948i$
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  + (−0.999 − 0.00650i)2-s + (0.945 + 0.325i)3-s + (0.999 + 0.0130i)4-s + (−0.100 + 0.994i)5-s + (−0.943 − 0.331i)6-s + (0.454 − 0.890i)7-s + (−0.999 − 0.0195i)8-s + (0.787 + 0.615i)9-s + (0.107 − 0.994i)10-s + (0.00975 + 0.999i)11-s + (0.941 + 0.337i)12-s + (0.870 − 0.491i)13-s + (−0.460 + 0.887i)14-s + (−0.419 + 0.907i)15-s + (0.999 + 0.0260i)16-s + (−0.998 + 0.0585i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.00650i)2-s + (0.945 + 0.325i)3-s + (0.999 + 0.0130i)4-s + (−0.100 + 0.994i)5-s + (−0.943 − 0.331i)6-s + (0.454 − 0.890i)7-s + (−0.999 − 0.0195i)8-s + (0.787 + 0.615i)9-s + (0.107 − 0.994i)10-s + (0.00975 + 0.999i)11-s + (0.941 + 0.337i)12-s + (0.870 − 0.491i)13-s + (−0.460 + 0.887i)14-s + (−0.419 + 0.907i)15-s + (0.999 + 0.0260i)16-s + (−0.998 + 0.0585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.315 + 0.948i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.315 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.930447927 + 1.392171268i\)
\(L(\frac12)\) \(\approx\) \(1.930447927 + 1.392171268i\)
\(L(1)\) \(\approx\) \(1.105179260 + 0.3351922999i\)
\(L(1)\) \(\approx\) \(1.105179260 + 0.3351922999i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.999 - 0.00650i)T \)
3 \( 1 + (0.945 + 0.325i)T \)
5 \( 1 + (-0.100 + 0.994i)T \)
7 \( 1 + (0.454 - 0.890i)T \)
11 \( 1 + (0.00975 + 0.999i)T \)
13 \( 1 + (0.870 - 0.491i)T \)
17 \( 1 + (-0.998 + 0.0585i)T \)
19 \( 1 + (0.964 - 0.263i)T \)
23 \( 1 + (0.724 + 0.689i)T \)
29 \( 1 + (0.977 - 0.212i)T \)
31 \( 1 + (0.985 + 0.168i)T \)
37 \( 1 + (-0.571 + 0.820i)T \)
41 \( 1 + (0.917 - 0.398i)T \)
43 \( 1 + (-0.692 - 0.721i)T \)
47 \( 1 + (-0.771 + 0.636i)T \)
53 \( 1 + (0.377 - 0.926i)T \)
59 \( 1 + (-0.741 - 0.670i)T \)
61 \( 1 + (0.803 + 0.595i)T \)
67 \( 1 + (0.638 + 0.769i)T \)
71 \( 1 + (0.494 - 0.869i)T \)
73 \( 1 + (0.377 + 0.926i)T \)
79 \( 1 + (-0.818 + 0.574i)T \)
83 \( 1 + (0.847 - 0.530i)T \)
89 \( 1 + (0.347 - 0.937i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−21.17917838818832204464380956002, −20.49734561825063706380204072437, −19.70698733433214001466996420038, −19.06960226505587907097493051055, −18.30735152670646544939310690217, −17.75443247014498765921135470750, −16.55453986565121565566037170187, −15.902107103869352190529499884732, −15.36369513662750595428333030187, −14.2522783552166783455523972667, −13.44231362368536383864448402493, −12.46905190417956641470989224730, −11.72161490580004325995054482188, −10.94210619767460471370295079330, −9.62065224391849079495438651249, −8.91531632328186834311961254666, −8.50861961046431427470990685175, −7.95097793860496988319799439582, −6.71567919601770869750843538620, −5.903392464471140108723512763448, −4.67176330430972126956120282882, −3.383483643486960325899951867131, −2.45357027517354482095889351525, −1.468080946620992385885267626, −0.72507892801609912619366068508, 1.02394532467267374135830464977, 2.03650760859622395653836374391, 2.98855181294950091737835792898, 3.76222696724379241049611304197, 4.93375316272611523258315784092, 6.57490088943539611555215407433, 7.14324487899678083757637240788, 7.869782491936574112112164032, 8.608983937704677319802955035952, 9.70170427305950842580779736317, 10.23602057785665351823481053216, 10.94579611678598459866949035068, 11.69595624269344617160984501747, 13.12954880648152708874771347063, 13.90915245499919824516841277431, 14.74614098117328935297056245246, 15.57444467864976286048660390302, 15.846942492733499316101250785656, 17.351300303533932917333832786649, 17.76484149348587763002089982309, 18.56551875100155596054068156463, 19.49161869748521062492964084249, 19.99771935845684633205261130090, 20.696621751074623166373229449713, 21.33107019781737017238364985120

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