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Properties

Label 1-195-195.98-r1-0-0
Degree $1$
Conductor $195$
Sign $0.503 + 0.863i$
Analytic cond. $20.9556$
Root an. cond. $20.9556$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s − 8-s + (0.866 − 0.5i)11-s + 14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s + (0.866 + 0.5i)22-s + (0.866 − 0.5i)23-s + (0.5 + 0.866i)28-s + (−0.5 − 0.866i)29-s + i·31-s + (0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)7-s − 8-s + (0.866 − 0.5i)11-s + 14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s + (0.866 + 0.5i)22-s + (0.866 − 0.5i)23-s + (0.5 + 0.866i)28-s + (−0.5 − 0.866i)29-s + i·31-s + (0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.503 + 0.863i$
Analytic conductor: \(20.9556\)
Root analytic conductor: \(20.9556\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (1:\ ),\ 0.503 + 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.246036544 + 1.290490156i\)
\(L(\frac12)\) \(\approx\) \(2.246036544 + 1.290490156i\)
\(L(1)\) \(\approx\) \(1.396534321 + 0.6088008789i\)
\(L(1)\) \(\approx\) \(1.396534321 + 0.6088008789i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + iT \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−27.02124786720542954127702347943, −25.458081996315638482949074068690, −24.60630427165058098297984200014, −23.63466442518425859305611484722, −22.53627747588412358984613583837, −21.90242851917171144177592790086, −20.88669234404469324720714263170, −20.107239332006770068929792087551, −18.99159100868964767540625739650, −18.24722623019443930360983538504, −17.1649475698308456806797075288, −15.59465289418151099349103068601, −14.71985082836730696494645043816, −13.89125612711485866744374309530, −12.60713616329262382469373613891, −11.83463055414629389353789498107, −11.03534230436214160149854298832, −9.592427314282505626450531707939, −8.97183213961255713524675716327, −7.316197532602171192448270118387, −5.77036722012671186990204716497, −4.92737616042220886724036314484, −3.59242814379317705216657098080, −2.347376632800316037979251375, −1.08704864168957885199780931761, 1.06489920590264814931381422099, 3.27222900510355225962431993910, 4.2446806107946497491594640555, 5.44448616727471423131011403114, 6.59788642632834290998740842712, 7.61842310220700965260456352421, 8.55115499668252464706243184667, 9.85141012125763667009817175660, 11.27679458016315278889096706831, 12.30896670491087916783261063367, 13.53943092859205680432806425724, 14.2682200948620939403520678651, 15.07633006057935101166533075547, 16.502200912138032454786131381096, 16.918647517879019252126631451823, 17.99345653647495012680074590821, 19.172887923963610604289807795647, 20.504458396201439368319414290089, 21.32850477994934140831362219164, 22.4258484494265936811292885186, 23.21381472723934521429469752854, 24.13470649268072823379585101936, 24.87250803929000233910135819751, 25.85907997762923299597409826087, 26.98559734783545270387536777020

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