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Properties

Label 1-967-967.438-r1-0-0
Degree $1$
Conductor $967$
Sign $0.917 + 0.397i$
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 + 0.0390i)2-s + (0.407 − 0.913i)3-s + (0.996 + 0.0779i)4-s + (0.822 + 0.568i)5-s + (0.442 − 0.896i)6-s + (−0.951 + 0.307i)7-s + (0.993 + 0.116i)8-s + (−0.668 − 0.744i)9-s + (0.799 + 0.600i)10-s + (−0.998 + 0.0585i)11-s + (0.477 − 0.878i)12-s + (0.998 + 0.0585i)13-s + (−0.962 + 0.269i)14-s + (0.854 − 0.519i)15-s + (0.987 + 0.155i)16-s + (0.938 − 0.344i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.0390i)2-s + (0.407 − 0.913i)3-s + (0.996 + 0.0779i)4-s + (0.822 + 0.568i)5-s + (0.442 − 0.896i)6-s + (−0.951 + 0.307i)7-s + (0.993 + 0.116i)8-s + (−0.668 − 0.744i)9-s + (0.799 + 0.600i)10-s + (−0.998 + 0.0585i)11-s + (0.477 − 0.878i)12-s + (0.998 + 0.0585i)13-s + (−0.962 + 0.269i)14-s + (0.854 − 0.519i)15-s + (0.987 + 0.155i)16-s + (0.938 − 0.344i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.917 + 0.397i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (438, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.917 + 0.397i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.346904196 + 1.109695955i\)
\(L(\frac12)\) \(\approx\) \(5.346904196 + 1.109695955i\)
\(L(1)\) \(\approx\) \(2.450481961 - 0.04172155670i\)
\(L(1)\) \(\approx\) \(2.450481961 - 0.04172155670i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.999 + 0.0390i)T \)
3 \( 1 + (0.407 - 0.913i)T \)
5 \( 1 + (0.822 + 0.568i)T \)
7 \( 1 + (-0.951 + 0.307i)T \)
11 \( 1 + (-0.998 + 0.0585i)T \)
13 \( 1 + (0.998 + 0.0585i)T \)
17 \( 1 + (0.938 - 0.344i)T \)
19 \( 1 + (0.0292 + 0.999i)T \)
23 \( 1 + (0.145 + 0.989i)T \)
29 \( 1 + (-0.279 + 0.960i)T \)
31 \( 1 + (0.527 + 0.849i)T \)
37 \( 1 + (-0.874 - 0.485i)T \)
41 \( 1 + (0.775 + 0.631i)T \)
43 \( 1 + (-0.126 + 0.991i)T \)
47 \( 1 + (0.544 - 0.838i)T \)
53 \( 1 + (0.682 - 0.730i)T \)
59 \( 1 + (-0.297 - 0.954i)T \)
61 \( 1 + (-0.775 - 0.631i)T \)
67 \( 1 + (-0.527 + 0.849i)T \)
71 \( 1 + (0.999 - 0.0390i)T \)
73 \( 1 + (0.682 + 0.730i)T \)
79 \( 1 + (0.864 - 0.502i)T \)
83 \( 1 + (-0.977 + 0.212i)T \)
89 \( 1 + (-0.527 + 0.849i)T \)
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

−21.165081978151273474529210918226, −21.0530446686863121385927696531, −20.33209155097398372960869496322, −19.46949927982609136524930178247, −18.55318585971494211232553243899, −17.018923275181825296012526578611, −16.66690178614276803549737986915, −15.65184868576643345618294233576, −15.41591067992330505022379533378, −14.087880299952374726647020310320, −13.56026817424851253380469188928, −13.03194528086656587543455386879, −12.09770226713903942309882695729, −10.7265353760232280604821309907, −10.40904980117978465221043926003, −9.49197895860534312387622243928, −8.549946043366374072366903294559, −7.51475245635742599926049661714, −6.18441871665460098509085451372, −5.689705274892500868733904293835, −4.74036552935772775476593893732, −3.92817143626019876035514356988, −2.96562454926670749103836878344, −2.28901283029734182233166262182, −0.73312267851530664988306229302, 1.24548318537227968608535718756, 2.14615493083690807264164843536, 3.17784608941233838316822395330, 3.439632914741998480757373130813, 5.38174477185321777270705869528, 5.83331562786506024535353401735, 6.661191986465375705609877684505, 7.364386561214000753640545682500, 8.32377057312613292219432245648, 9.55456573116847486420726945765, 10.39207284895938402787620811341, 11.335219971520735314346200406714, 12.43035550701556015865300973178, 12.872859723270817160903948538537, 13.69624210596430085723283896009, 14.121691541414643425552330424, 15.058565890803381952119122509478, 15.91748693875516661241209982758, 16.69610396663462231135244628013, 17.90723131322069021278085080394, 18.559239932520884213740396388656, 19.20887360556557067570486604042, 20.14227245373526695981468601303, 21.10436677781774643277069742326, 21.39778233488100578080940809836

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