Properties

Label 1-967-967.431-r0-0-0
Degree $1$
Conductor $967$
Sign $0.790 + 0.612i$
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  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)5-s + (0.623 − 0.781i)6-s + (−0.222 − 0.974i)7-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.222 − 0.974i)10-s + (−0.900 − 0.433i)11-s + (0.623 + 0.781i)12-s + (−0.900 + 0.433i)13-s + 14-s + 15-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 − 0.433i)4-s + (−0.900 + 0.433i)5-s + (0.623 − 0.781i)6-s + (−0.222 − 0.974i)7-s + (0.623 − 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.222 − 0.974i)10-s + (−0.900 − 0.433i)11-s + (0.623 + 0.781i)12-s + (−0.900 + 0.433i)13-s + 14-s + 15-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2331585980 + 0.07979698113i\)
\(L(\frac12)\) \(\approx\) \(0.2331585980 + 0.07979698113i\)
\(L(1)\) \(\approx\) \(0.3796815156 + 0.1150405025i\)
\(L(1)\) \(\approx\) \(0.3796815156 + 0.1150405025i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 + (-0.222 + 0.974i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + T \)
79 \( 1 + (0.623 + 0.781i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
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

−21.92039866638671745984867078468, −20.74150609843265189464724651120, −20.23254589908981366442044550749, −19.37562079378985908534978214919, −18.43504829130551094010591320091, −17.953026257583339809282841591475, −17.06991297813061243807774281723, −16.15061305578245206411391607903, −15.46999881700391960566603461943, −14.76617399658883191541129662937, −13.021850557072482128824172862945, −12.74524140630332921353593821428, −11.87763873488728777774065418088, −11.34601409037044113188483153115, −10.51221536447904933765802153865, −9.59660865224140243272491747076, −8.93988133722434416168928607283, −7.93671394797300220661585531555, −6.96888348731658341840388710204, −5.46839769023785413586944816903, −4.90555961437434371539347662631, −4.1644864360725328125558606653, −2.99142929453026857611436807881, −2.063515319022506036715101658689, −0.37947398301058140274430915852, 0.36543445488389590160684310854, 1.92924942571977212386443919476, 3.74761344534519590629288668999, 4.38826688494755314492897819826, 5.44470226340276600445277602593, 6.29973622880505776527281958512, 7.22937751126135201127572469071, 7.54575248044480998334505131176, 8.41412101170832724651181495201, 9.84212586426908766552239543905, 10.548026628022084199205207614118, 11.21204024384027787897563850711, 12.35716254538518249407953468138, 13.11861320707653813212174118670, 13.93078297240040511214406459699, 14.8353630709247334198020314150, 15.72726573912675905239843088032, 16.48497688797407716964637868444, 16.8377085079148261415159431560, 17.97495661276153795132762407857, 18.368018053146468373640532394853, 19.458236704743725342337146609216, 19.69783808245616804585815530419, 21.322791293805658642352815188548, 22.26634242633621680957043160711

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