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Properties

Label 1-95-95.68-r1-0-0
Degree $1$
Conductor $95$
Sign $0.181 - 0.983i$
Analytic cond. $10.2091$
Root an. cond. $10.2091$
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  + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.5 + 0.866i)6-s − i·7-s + i·8-s + (0.5 − 0.866i)9-s + 11-s − i·12-s + (−0.866 − 0.5i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + i·18-s + (−0.5 − 0.866i)21-s + (−0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.5 + 0.866i)6-s − i·7-s + i·8-s + (0.5 − 0.866i)9-s + 11-s − i·12-s + (−0.866 − 0.5i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + i·18-s + (−0.5 − 0.866i)21-s + (−0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.181 - 0.983i$
Analytic conductor: \(10.2091\)
Root analytic conductor: \(10.2091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (1:\ ),\ 0.181 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.041142134 - 0.8662451690i\)
\(L(\frac12)\) \(\approx\) \(1.041142134 - 0.8662451690i\)
\(L(1)\) \(\approx\) \(0.9402354877 - 0.2417258547i\)
\(L(1)\) \(\approx\) \(0.9402354877 - 0.2417258547i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 - iT \)
11 \( 1 + T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−30.16736371316128081691000398922, −28.98263183830920296346751707308, −27.86734918650052519788617299208, −27.13897975495466723384652711777, −26.17941230941762434807912187677, −25.12884932057098035336453661534, −24.51284653726250089246289663418, −22.04602215395298411164033839859, −21.76575071284274352988220566082, −20.37920336945404285303658969712, −19.58078175490721987160648978433, −18.72084056031961689893179480870, −17.44471568210826989506990840276, −16.176196723131572332557185405555, −15.23228015382199225222368757101, −13.9466327459888202958469207020, −12.38029009299466137667697536292, −11.38208595758594926299295341485, −9.82485564408653036957333150029, −9.149564978000021479758756351065, −8.16758565937952582535001923971, −6.71843632266816918758461571771, −4.5136508696141008441956911837, −3.00333292326442203381306374836, −1.87618786572459044516312858117, 0.7313878049530479436133578229, 2.247274541621864851946992312898, 4.17080147034489981236778821093, 6.342377504800982694730409299295, 7.29626487198053056254298344372, 8.299856402782605917204502996980, 9.469699581590003278390308447550, 10.52154366521334479934888586588, 12.1438084855807381392769913486, 13.73232895194969039580621526994, 14.54547811418445544737771276401, 15.6542587414641576078573446018, 17.124383419330932018089211943832, 17.77633500486283755356536970929, 19.307480924138913961738699983594, 19.761162556551667736405529793012, 20.72133975943750661938077010459, 22.56517298793284200345882755468, 23.97068354071482142469619624117, 24.58251903097824974840998503895, 25.601850049418624949271981930589, 26.63213462443873932691340912741, 27.16522652859113344865013567497, 28.64851063809106547101404904710, 29.81270295766736064298987947253

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