Properties

Label 1-109-109.66-r0-0-0
Degree $1$
Conductor $109$
Sign $-0.635 + 0.771i$
Analytic cond. $0.506193$
Root an. cond. $0.506193$
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 + (−0.939 + 0.342i)3-s + (−0.5 − 0.866i)4-s + (0.766 + 0.642i)5-s + (0.173 − 0.984i)6-s + (0.766 + 0.642i)7-s + 8-s + (0.766 − 0.642i)9-s + (−0.939 + 0.342i)10-s + (−0.939 − 0.342i)11-s + (0.766 + 0.642i)12-s + (0.766 + 0.642i)13-s + (−0.939 + 0.342i)14-s + (−0.939 − 0.342i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.939 + 0.342i)3-s + (−0.5 − 0.866i)4-s + (0.766 + 0.642i)5-s + (0.173 − 0.984i)6-s + (0.766 + 0.642i)7-s + 8-s + (0.766 − 0.642i)9-s + (−0.939 + 0.342i)10-s + (−0.939 − 0.342i)11-s + (0.766 + 0.642i)12-s + (0.766 + 0.642i)13-s + (−0.939 + 0.342i)14-s + (−0.939 − 0.342i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(109\)
Sign: $-0.635 + 0.771i$
Analytic conductor: \(0.506193\)
Root analytic conductor: \(0.506193\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{109} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 109,\ (0:\ ),\ -0.635 + 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2765527964 + 0.5862896350i\)
\(L(\frac12)\) \(\approx\) \(0.2765527964 + 0.5862896350i\)
\(L(1)\) \(\approx\) \(0.5389659608 + 0.4499989115i\)
\(L(1)\) \(\approx\) \(0.5389659608 + 0.4499989115i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad109 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (-0.939 - 0.342i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−29.11904278478793952973589128726, −28.217677547731793439151192289186, −27.611645950759158648665153482894, −26.307464786235414792037629295836, −25.12256466557562039825609559909, −23.8837827309609938115889958946, −22.935061281633930802241828418050, −21.68173362288525826274972724701, −20.86547821946965253093043521994, −19.993310640527558795962572825502, −18.331165068541434112604283385976, −17.78109140921040908847594776659, −17.04798121449552243086094558426, −15.84651495929756551851802604554, −13.48234789459324837010735411556, −13.17125265699876432316981332754, −11.75856282131268468170269399375, −10.80165048932310439251860126783, −9.92585573076427614137069270656, −8.41773812650488191666473039267, −7.245169686665294055037251500122, −5.42133123792093304930161191241, −4.43850279938503681249889117100, −2.20980555517812435414564118860, −0.90656889947459384939069093362, 1.83042054728573643622946530765, 4.43236359668207022457015263325, 5.831061532520352404948152708451, 6.24875120598870932974401988156, 7.90899182877942802122561387308, 9.21341848683905149270346322317, 10.49681898649020215766564640083, 11.11963286972425825099528876977, 12.951535716617286778669294295968, 14.3463242471875335727829753969, 15.26133225800564800754776021468, 16.34004261732298106611132163398, 17.35015695713018451751713154701, 18.31536367814734354812366302575, 18.704680047849815155216530728299, 20.95169309026700779317056217939, 21.75684245469813127046186685369, 22.862825527951844568026514669581, 23.8491481463301501360658238371, 24.71368492372426773644331036903, 26.0769390520688435203220596584, 26.632188381899865807192558325067, 27.98131516748405354465964021988, 28.506102771961265263759216911158, 29.58073759498631872918769457301

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