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Properties

Label 1-967-967.101-r0-0-0
Degree $1$
Conductor $967$
Sign $0.372 - 0.928i$
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  + (0.847 − 0.530i)2-s + (−0.967 + 0.250i)3-s + (0.436 − 0.899i)4-s + (0.728 + 0.684i)5-s + (−0.687 + 0.726i)6-s + (0.959 + 0.282i)7-s + (−0.107 − 0.994i)8-s + (0.874 − 0.485i)9-s + (0.981 + 0.193i)10-s + (−0.668 − 0.744i)11-s + (−0.197 + 0.980i)12-s + (0.978 + 0.206i)13-s + (0.962 − 0.269i)14-s + (−0.877 − 0.480i)15-s + (−0.618 − 0.785i)16-s + (0.316 − 0.948i)17-s + ⋯
L(s)  = 1  + (0.847 − 0.530i)2-s + (−0.967 + 0.250i)3-s + (0.436 − 0.899i)4-s + (0.728 + 0.684i)5-s + (−0.687 + 0.726i)6-s + (0.959 + 0.282i)7-s + (−0.107 − 0.994i)8-s + (0.874 − 0.485i)9-s + (0.981 + 0.193i)10-s + (−0.668 − 0.744i)11-s + (−0.197 + 0.980i)12-s + (0.978 + 0.206i)13-s + (0.962 − 0.269i)14-s + (−0.877 − 0.480i)15-s + (−0.618 − 0.785i)16-s + (0.316 − 0.948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.372 - 0.928i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.372 - 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.889468821 - 1.278041775i\)
\(L(\frac12)\) \(\approx\) \(1.889468821 - 1.278041775i\)
\(L(1)\) \(\approx\) \(1.503806968 - 0.5106260172i\)
\(L(1)\) \(\approx\) \(1.503806968 - 0.5106260172i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.847 - 0.530i)T \)
3 \( 1 + (-0.967 + 0.250i)T \)
5 \( 1 + (0.728 + 0.684i)T \)
7 \( 1 + (0.959 + 0.282i)T \)
11 \( 1 + (-0.668 - 0.744i)T \)
13 \( 1 + (0.978 + 0.206i)T \)
17 \( 1 + (0.316 - 0.948i)T \)
19 \( 1 + (-0.587 - 0.809i)T \)
23 \( 1 + (-0.864 - 0.502i)T \)
29 \( 1 + (0.924 - 0.380i)T \)
31 \( 1 + (-0.395 - 0.918i)T \)
37 \( 1 + (0.448 + 0.893i)T \)
41 \( 1 + (-0.775 - 0.631i)T \)
43 \( 1 + (0.969 - 0.244i)T \)
47 \( 1 + (-0.922 + 0.386i)T \)
53 \( 1 + (0.291 + 0.956i)T \)
59 \( 1 + (0.929 - 0.368i)T \)
61 \( 1 + (-0.158 + 0.987i)T \)
67 \( 1 + (0.993 - 0.116i)T \)
71 \( 1 + (-0.883 + 0.468i)T \)
73 \( 1 + (0.291 - 0.956i)T \)
79 \( 1 + (-0.677 + 0.735i)T \)
83 \( 1 + (-0.555 - 0.831i)T \)
89 \( 1 + (-0.597 - 0.801i)T \)
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

−21.78120812495763328834425932215, −21.2817479005236322598260759336, −20.7812587509424978184105680772, −19.79746428861197974627425981948, −18.14260395895006949980983699983, −17.86079591352724725655183638449, −17.13316887440269091740115648498, −16.36296301373240272473163660971, −15.73681195666831578193292149100, −14.64876620124614218213509006116, −13.89328092922766009763977585362, −12.92772569579439057973229257782, −12.61108347757638935508008498792, −11.67758689332541352307344340126, −10.73137103173849232862667486735, −10.07359290262124897795681011178, −8.422801083380369575709639679157, −7.97169512476231388018976082260, −6.85756151075216869984453550236, −5.94961969739656768010704776965, −5.39765462573702443764761905473, −4.63223280941157109184195102746, −3.83226922538074510309557931282, −2.055799378283112898188575766349, −1.42609201150692201013040244599, 0.88720251574293909334268518088, 2.04915770060520982132441932906, 2.94132202436547725806467739258, 4.16382406493318249375964119878, 4.98958452994749900823163316885, 5.81186202909720238276610860814, 6.283898590281748416169445238354, 7.33403392926979228042225838803, 8.74010498508207511512881354829, 9.918105492836829129006497925258, 10.56480596967152844422990807733, 11.313363791636411357413652482661, 11.63869761062513915448799328760, 12.84495632519369568736677485329, 13.649685133856373924407291289468, 14.2425219290355469389423961046, 15.277521233305596468446958364835, 15.84699448400782789572999661978, 16.8370735726196980969948603215, 17.95598137749562773862040189164, 18.38596743065180550737679241799, 19.02523232968909462156698184305, 20.59377123820269428485174230607, 20.99654660266106393484531309324, 21.68843607292834537480978307313

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