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Properties

Label 1-967-967.156-r0-0-0
Degree $1$
Conductor $967$
Sign $0.0716 + 0.997i$
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  + (0.177 + 0.984i)2-s + (−0.297 + 0.954i)3-s + (−0.936 + 0.350i)4-s + (0.914 − 0.404i)5-s + (−0.992 − 0.123i)6-s + (−0.927 − 0.374i)7-s + (−0.511 − 0.859i)8-s + (−0.822 − 0.568i)9-s + (0.560 + 0.828i)10-s + (0.494 + 0.869i)11-s + (−0.0552 − 0.998i)12-s + (−0.999 + 0.00650i)13-s + (0.203 − 0.979i)14-s + (0.113 + 0.993i)15-s + (0.754 − 0.655i)16-s + (0.999 + 0.0390i)17-s + ⋯
L(s)  = 1  + (0.177 + 0.984i)2-s + (−0.297 + 0.954i)3-s + (−0.936 + 0.350i)4-s + (0.914 − 0.404i)5-s + (−0.992 − 0.123i)6-s + (−0.927 − 0.374i)7-s + (−0.511 − 0.859i)8-s + (−0.822 − 0.568i)9-s + (0.560 + 0.828i)10-s + (0.494 + 0.869i)11-s + (−0.0552 − 0.998i)12-s + (−0.999 + 0.00650i)13-s + (0.203 − 0.979i)14-s + (0.113 + 0.993i)15-s + (0.754 − 0.655i)16-s + (0.999 + 0.0390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0716 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0716 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.0716 + 0.997i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.0716 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9672526156 + 0.9002887681i\)
\(L(\frac12)\) \(\approx\) \(0.9672526156 + 0.9002887681i\)
\(L(1)\) \(\approx\) \(0.8251461576 + 0.6105019402i\)
\(L(1)\) \(\approx\) \(0.8251461576 + 0.6105019402i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.177 + 0.984i)T \)
3 \( 1 + (-0.297 + 0.954i)T \)
5 \( 1 + (0.914 - 0.404i)T \)
7 \( 1 + (-0.927 - 0.374i)T \)
11 \( 1 + (0.494 + 0.869i)T \)
13 \( 1 + (-0.999 + 0.00650i)T \)
17 \( 1 + (0.999 + 0.0390i)T \)
19 \( 1 + (0.867 - 0.497i)T \)
23 \( 1 + (0.874 - 0.485i)T \)
29 \( 1 + (-0.371 - 0.928i)T \)
31 \( 1 + (0.0617 - 0.998i)T \)
37 \( 1 + (0.228 + 0.973i)T \)
41 \( 1 + (0.962 + 0.269i)T \)
43 \( 1 + (-0.982 + 0.187i)T \)
47 \( 1 + (-0.993 - 0.110i)T \)
53 \( 1 + (-0.419 - 0.907i)T \)
59 \( 1 + (0.978 + 0.206i)T \)
61 \( 1 + (-0.247 - 0.968i)T \)
67 \( 1 + (0.833 - 0.552i)T \)
71 \( 1 + (0.763 + 0.646i)T \)
73 \( 1 + (-0.419 + 0.907i)T \)
79 \( 1 + (0.448 + 0.893i)T \)
83 \( 1 + (0.947 - 0.319i)T \)
89 \( 1 + (-0.895 - 0.445i)T \)
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

−21.744390488618047429613563313985, −20.86827797300107847815514283302, −19.603416056880470111870651820911, −19.345216128793350664337339864490, −18.51197174009143154429926252133, −17.966534376279064607331665894535, −17.02863650022805093004909915764, −16.3766161818125759596742677091, −14.67253415433221958820955696520, −14.18829839514346850707606825661, −13.41849212958332804220286688487, −12.66448909223650205533147148131, −12.099944927947831305417223816211, −11.21584849127959944761284632912, −10.331311532671053049256913843352, −9.49670860640995309646377186216, −8.85689805246661141368616098733, −7.53707327088804665866916258950, −6.58070820651067154768202444317, −5.640198879411981186995724809582, −5.249517877750656910556341723057, −3.32610842638937345292538061815, −2.95159013145367593635663492222, −1.84236361556040515820493020304, −0.929973022388308891617955865158, 0.759029896432151118327471452907, 2.69529111841344034328555719143, 3.72020047620607146026556805672, 4.73654654772493322012046301956, 5.244177845677189679011375029471, 6.24341591730806691949436365839, 6.88786825731074130294558443483, 8.0077207421925197848840091633, 9.27896578572121516370153931092, 9.73725448932512380947725328781, 10.00798007570414252438898330617, 11.62457973768386947114142660465, 12.58902571794355220241576690724, 13.239741760229824027651954394548, 14.276518486788087532236312140356, 14.82363062746250545437542753668, 15.67776501211829148741204043451, 16.60899274003829511385167560559, 16.97075795871837282741840424320, 17.498644566630451655170323385611, 18.55075081201472965060197869165, 19.697572588286688928170189060033, 20.57290553439230379965155087905, 21.344770308232396935090278005056, 22.23400641695608666128666399000

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