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Properties

Label 1-117-117.110-r0-0-0
Degree $1$
Conductor $117$
Sign $0.952 + 0.305i$
Analytic cond. $0.543345$
Root an. cond. $0.543345$
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 + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + i·7-s − i·8-s + (0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s − i·20-s + (−0.5 − 0.866i)22-s + 23-s + (0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + i·7-s − i·8-s + (0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s − i·20-s + (−0.5 − 0.866i)22-s + 23-s + (0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.952 + 0.305i$
Analytic conductor: \(0.543345\)
Root analytic conductor: \(0.543345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (0:\ ),\ 0.952 + 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6097151304 + 0.09541121980i\)
\(L(\frac12)\) \(\approx\) \(0.6097151304 + 0.09541121980i\)
\(L(1)\) \(\approx\) \(0.6659085684 + 0.01812298310i\)
\(L(1)\) \(\approx\) \(0.6659085684 + 0.01812298310i\)

Euler product

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

−29.11783234602121659237004790661, −27.783698698199183247804266442010, −26.876421609609381191125651749364, −26.56234234769849124719504825321, −25.16959362446638525517574857942, −24.157084695354564311263670718, −23.273975153403488858832193914277, −22.30515441311615030657687307415, −20.448313216120179497229098019875, −19.734010290974155775426237408923, −18.84492815114390041860636665850, −17.71370460576671670601690332884, −16.649529503472134532078457205972, −15.79214957445637346570972383305, −14.649656629177603400488570818840, −13.683575383397319227491988424887, −11.63677570167886252333917460778, −10.98436100907993124206315741654, −9.71341914246516039773153384336, −8.46804424371464918387631553676, −7.26280442863311733573989314731, −6.62369966009090637940237394090, −4.73856718728446611510605401659, −3.13200009455603255490952542681, −0.90360310413768660320602497964, 1.470836900818648603480513859108, 3.126606160340175363150960216070, 4.52350064755463807081016286829, 6.441162056626149070507594049913, 7.86232130661416179267335564940, 8.767027023739913690293585000904, 9.7219415762604806025881014652, 11.31125637712081166911432239008, 12.02280150262734112948390054003, 12.91112399201234958335455337528, 14.932677948555566979070382611478, 15.83480974093782338799396895476, 16.90455498071493264772862073886, 17.9654700775053432098369799196, 19.13956087671786475707089229210, 19.75901200570919929988524303925, 20.85257661148289001444012767309, 21.90110438956894177758770414299, 23.03046535865268135772117541865, 24.64239931826080157680762831319, 25.09445860542703143753469297933, 26.55675902683610324546969516642, 27.35082337663416254877980143746, 28.31297033443049086022366364991, 28.77707118021407782082172749194

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