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Properties

Label 1-117-117.112-r1-0-0
Degree $1$
Conductor $117$
Sign $0.992 - 0.119i$
Analytic cond. $12.5733$
Root an. cond. $12.5733$
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 + (0.866 + 0.5i)7-s − i·8-s − 10-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s + i·19-s + (0.866 + 0.5i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)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 + (0.866 + 0.5i)7-s − i·8-s − 10-s + (0.866 + 0.5i)11-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s + i·19-s + (0.866 + 0.5i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.992 - 0.119i$
Analytic conductor: \(12.5733\)
Root analytic conductor: \(12.5733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (1:\ ),\ 0.992 - 0.119i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.542177034 - 0.09223041749i\)
\(L(\frac12)\) \(\approx\) \(1.542177034 - 0.09223041749i\)
\(L(1)\) \(\approx\) \(0.9989434147 - 0.1199131863i\)
\(L(1)\) \(\approx\) \(0.9989434147 - 0.1199131863i\)

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 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 - T \)
19 \( 1 + iT \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + iT \)
73 \( 1 - iT \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 - iT \)
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

−28.86454411142712779224807083702, −27.82019979154595338662271718058, −26.70548148714183554970071826956, −26.22777327288930342165505879519, −24.77642311662490606525153031299, −24.4253476866177275374515798011, −22.99817764323530047825971643211, −21.75735671782511242195417708964, −20.609765944455044515426148431656, −19.53277838340167970213040649503, −18.38412050699122843287051439504, −17.460467813899058270189189140681, −16.89685329266238191689311542792, −15.36207866313268450083619094989, −14.38314515607438349039964925194, −13.54152368582290158776008021246, −11.40802714320822569626473615617, −10.708793850888000323671275592929, −9.45309639564862068008044976329, −8.458206784294944564884435723278, −7.03536720538891541512996747314, −6.19216359111387618734979224081, −4.70205602758007801626685230244, −2.43914604212502624210420991055, −1.030279022923650014893463790583, 1.318674009847535593080296416252, 2.27180116648421790900720208946, 4.22178503075494060460430923909, 5.81869241894745024751388440236, 7.31462955667543751540303281141, 8.725557967783267906685282041348, 9.35640013352861834868847287596, 10.652885943678055716385583834446, 11.80911379962000933708493133655, 12.75996045266270795067613356525, 14.14158905927936922723949442396, 15.505505938790004781274510005172, 16.91413079470387658361471934485, 17.55721095872910125031911937539, 18.42621676498001024565406529761, 19.734415718779805880463489486156, 20.71194546541140691150681838227, 21.447736346447568316079331403381, 22.45204988975433514839019037220, 24.365104471286460579698029933623, 24.98258584770049305654847167134, 25.8841816620200958627871661747, 27.22483718006249673368333757417, 27.898115403808826552702330342520, 28.8387444476771474445302583675

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