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Properties

Label 1-967-967.110-r1-0-0
Degree $1$
Conductor $967$
Sign $0.535 - 0.844i$
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  + (0.996 − 0.0779i)2-s + (0.668 − 0.744i)3-s + (0.987 − 0.155i)4-s + (−0.353 + 0.935i)5-s + (0.608 − 0.793i)6-s + (−0.811 − 0.584i)7-s + (0.972 − 0.232i)8-s + (−0.107 − 0.994i)9-s + (−0.279 + 0.960i)10-s + (0.993 + 0.116i)11-s + (0.544 − 0.838i)12-s + (−0.993 + 0.116i)13-s + (−0.854 − 0.519i)14-s + (0.460 + 0.887i)15-s + (0.951 − 0.307i)16-s + (0.763 + 0.646i)17-s + ⋯
L(s)  = 1  + (0.996 − 0.0779i)2-s + (0.668 − 0.744i)3-s + (0.987 − 0.155i)4-s + (−0.353 + 0.935i)5-s + (0.608 − 0.793i)6-s + (−0.811 − 0.584i)7-s + (0.972 − 0.232i)8-s + (−0.107 − 0.994i)9-s + (−0.279 + 0.960i)10-s + (0.993 + 0.116i)11-s + (0.544 − 0.838i)12-s + (−0.993 + 0.116i)13-s + (−0.854 − 0.519i)14-s + (0.460 + 0.887i)15-s + (0.951 − 0.307i)16-s + (0.763 + 0.646i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.752686784 - 2.614739601i\)
\(L(\frac12)\) \(\approx\) \(4.752686784 - 2.614739601i\)
\(L(1)\) \(\approx\) \(2.328140596 - 0.6382386635i\)
\(L(1)\) \(\approx\) \(2.328140596 - 0.6382386635i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.996 - 0.0779i)T \)
3 \( 1 + (0.668 - 0.744i)T \)
5 \( 1 + (-0.353 + 0.935i)T \)
7 \( 1 + (-0.811 - 0.584i)T \)
11 \( 1 + (0.993 + 0.116i)T \)
13 \( 1 + (-0.993 + 0.116i)T \)
17 \( 1 + (0.763 + 0.646i)T \)
19 \( 1 + (0.998 + 0.0585i)T \)
23 \( 1 + (0.957 + 0.288i)T \)
29 \( 1 + (0.844 - 0.536i)T \)
31 \( 1 + (-0.442 - 0.896i)T \)
37 \( 1 + (-0.527 + 0.849i)T \)
41 \( 1 + (-0.203 + 0.979i)T \)
43 \( 1 + (0.967 - 0.250i)T \)
47 \( 1 + (0.407 - 0.913i)T \)
53 \( 1 + (-0.0682 + 0.997i)T \)
59 \( 1 + (-0.822 - 0.568i)T \)
61 \( 1 + (0.203 - 0.979i)T \)
67 \( 1 + (0.442 - 0.896i)T \)
71 \( 1 + (0.996 + 0.0779i)T \)
73 \( 1 + (-0.0682 - 0.997i)T \)
79 \( 1 + (-0.494 - 0.869i)T \)
83 \( 1 + (0.909 + 0.416i)T \)
89 \( 1 + (0.442 - 0.896i)T \)
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

−21.67354767300275207055489609780, −20.99140451457615802958754817053, −20.20389070206966908052603831274, −19.5868682346084438101926902452, −19.111187238345362324241024705315, −17.355445057281241016790010743578, −16.43886460080651673368674653870, −16.08607278344910053912299172370, −15.36462245378323045461047724702, −14.42096999883474276114255693439, −13.95513676622904345797010977012, −12.78449165514980248313245296431, −12.28387925091467886555128212697, −11.52615605186440783913830132233, −10.32884468922337259493860277632, −9.36683079984933654538897849215, −8.85204792482612561532036600201, −7.65043400879227003903130060758, −6.887284588000686591131293915336, −5.44177682193868045153420603075, −5.09221878194571494554000784372, −4.02491893251070608667880967145, −3.27095861951934870264391373212, −2.50371932747363854146968946146, −1.08476452389829842585941658069, 0.81043818756469818175555196737, 1.9733982942422918281318223917, 3.11648691764412425146970300935, 3.423655206052903405464606223422, 4.44621259171089132209944165416, 5.94322848474088510310098547600, 6.66493195477958487332334019761, 7.26386405121198209375600190897, 7.87395144939480619618832376059, 9.46689225688211147403910010929, 10.10078776065051572348494247583, 11.250463495026584041550700985112, 12.07941151250572672487920399473, 12.590720532589602781094249858209, 13.736234308403109877092423728306, 14.06006246537456828571224683452, 14.93788816438561840980298661472, 15.41584743221658813205447526276, 16.69228901437007347431678309354, 17.34688662607895561704945579733, 18.756981581070336418107288468376, 19.249334922095592010518822569278, 19.83684170759610531763958880580, 20.45510313175644864039585659832, 21.64865319011680917278169766839

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