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Properties

Label 1-967-967.52-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.655 + 0.754i$
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  + (−0.974 + 0.225i)2-s + (0.576 + 0.816i)3-s + (0.898 − 0.439i)4-s + (−0.377 − 0.926i)5-s + (−0.746 − 0.665i)6-s + (0.715 + 0.699i)7-s + (−0.775 + 0.631i)8-s + (−0.334 + 0.942i)9-s + (0.576 + 0.816i)10-s + (−0.334 + 0.942i)11-s + (0.877 + 0.480i)12-s + (0.648 − 0.761i)13-s + (−0.854 − 0.519i)14-s + (0.538 − 0.842i)15-s + (0.613 − 0.789i)16-s + (0.460 − 0.887i)17-s + ⋯
L(s)  = 1  + (−0.974 + 0.225i)2-s + (0.576 + 0.816i)3-s + (0.898 − 0.439i)4-s + (−0.377 − 0.926i)5-s + (−0.746 − 0.665i)6-s + (0.715 + 0.699i)7-s + (−0.775 + 0.631i)8-s + (−0.334 + 0.942i)9-s + (0.576 + 0.816i)10-s + (−0.334 + 0.942i)11-s + (0.877 + 0.480i)12-s + (0.648 − 0.761i)13-s + (−0.854 − 0.519i)14-s + (0.538 − 0.842i)15-s + (0.613 − 0.789i)16-s + (0.460 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.655 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.655 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.655 + 0.754i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.655 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5319558444 + 1.166672623i\)
\(L(\frac12)\) \(\approx\) \(0.5319558444 + 1.166672623i\)
\(L(1)\) \(\approx\) \(0.7754513945 + 0.3234874626i\)
\(L(1)\) \(\approx\) \(0.7754513945 + 0.3234874626i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.974 + 0.225i)T \)
3 \( 1 + (0.576 + 0.816i)T \)
5 \( 1 + (-0.377 - 0.926i)T \)
7 \( 1 + (0.715 + 0.699i)T \)
11 \( 1 + (-0.334 + 0.942i)T \)
13 \( 1 + (0.648 - 0.761i)T \)
17 \( 1 + (0.460 - 0.887i)T \)
19 \( 1 + (-0.995 + 0.0909i)T \)
23 \( 1 + (0.0682 - 0.997i)T \)
29 \( 1 + (0.334 + 0.942i)T \)
31 \( 1 + (0.934 + 0.356i)T \)
37 \( 1 + (0.648 - 0.761i)T \)
41 \( 1 + (-0.203 + 0.979i)T \)
43 \( 1 + (0.998 + 0.0455i)T \)
47 \( 1 + (-0.538 + 0.842i)T \)
53 \( 1 + (-0.829 - 0.557i)T \)
59 \( 1 + (-0.829 + 0.557i)T \)
61 \( 1 + (-0.949 + 0.313i)T \)
67 \( 1 + (0.775 + 0.631i)T \)
71 \( 1 + (0.682 - 0.730i)T \)
73 \( 1 + (-0.829 + 0.557i)T \)
79 \( 1 + (0.829 - 0.557i)T \)
83 \( 1 + (0.746 + 0.665i)T \)
89 \( 1 + (0.158 - 0.987i)T \)
97 \( 1 + T \)
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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

−21.183378789663784177196850593815, −20.277123891369866952013973158886, −19.20313894222628941647737616740, −19.130484810144238189181468587166, −18.35192338808564801710579911000, −17.4476790074449639661094467840, −16.91750712266540509169890118735, −15.64839070796884784554128959980, −15.00475374690659124377135712740, −14.01465091801952407278301849301, −13.44102198457725751623806963388, −12.21653287553013906987964677266, −11.3607916048177674434000027594, −10.904257157660747817789188086750, −9.97338117982419228341965017245, −8.812762599974460920525670686915, −8.0041350005422758558128947225, −7.71386575578177036592072479476, −6.56816777532613253331894449021, −6.12822365629244409386069098254, −4.02209237466385617352407546514, −3.33759346902201944632576302787, −2.29457363871503219426888798856, −1.426603055909128783816643352413, −0.38482911181938222689830452747, 1.03063687707575837074497174655, 2.157216786008764501989138285994, 3.03854331907718752949078568265, 4.554972362087866741102995517331, 5.05684644889701822914728777842, 6.121312897493227853700399508736, 7.62129521945474403164117500355, 8.13567966493078283219553728415, 8.80867239490320186996069125106, 9.452669781102968589382774009604, 10.40673150697462033985933708881, 11.102023246293208847464535905103, 12.11949928828807027243023059686, 12.886312040724334698071213854060, 14.3358049246652879005108379308, 14.938456158936077535583844059638, 15.732217154059595870093487942432, 16.13233072363893216165521218044, 17.07443958503875272665048214427, 17.8946163185846156227239598983, 18.64549111606667510619887594456, 19.6445403773664770657397845250, 20.27401135746459331859547921075, 20.94188240036393011136503053813, 21.23872454687245187755521189268

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