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Properties

Label 1-116-116.107-r1-0-0
Degree $1$
Conductor $116$
Sign $-0.695 - 0.718i$
Analytic cond. $12.4659$
Root an. cond. $12.4659$
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  + (0.222 − 0.974i)3-s + (0.623 − 0.781i)5-s + (0.222 − 0.974i)7-s + (−0.900 − 0.433i)9-s + (0.900 − 0.433i)11-s + (−0.900 + 0.433i)13-s + (−0.623 − 0.781i)15-s + 17-s + (0.222 + 0.974i)19-s + (−0.900 − 0.433i)21-s + (−0.623 − 0.781i)23-s + (−0.222 − 0.974i)25-s + (−0.623 + 0.781i)27-s + (−0.623 + 0.781i)31-s + (−0.222 − 0.974i)33-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)3-s + (0.623 − 0.781i)5-s + (0.222 − 0.974i)7-s + (−0.900 − 0.433i)9-s + (0.900 − 0.433i)11-s + (−0.900 + 0.433i)13-s + (−0.623 − 0.781i)15-s + 17-s + (0.222 + 0.974i)19-s + (−0.900 − 0.433i)21-s + (−0.623 − 0.781i)23-s + (−0.222 − 0.974i)25-s + (−0.623 + 0.781i)27-s + (−0.623 + 0.781i)31-s + (−0.222 − 0.974i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $-0.695 - 0.718i$
Analytic conductor: \(12.4659\)
Root analytic conductor: \(12.4659\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 116,\ (1:\ ),\ -0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7591330350 - 1.790394498i\)
\(L(\frac12)\) \(\approx\) \(0.7591330350 - 1.790394498i\)
\(L(1)\) \(\approx\) \(1.041049959 - 0.7618812111i\)
\(L(1)\) \(\approx\) \(1.041049959 - 0.7618812111i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
7 \( 1 + (0.222 - 0.974i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + T \)
19 \( 1 + (0.222 + 0.974i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (-0.623 + 0.781i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 + (0.900 + 0.433i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (0.900 + 0.433i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−29.39620793839911173491375867847, −28.01872420964171826616320294775, −27.51142366354421296531928083157, −26.23227151491965340960582505866, −25.49523428938331451676755056027, −24.607471135590545863631691441489, −22.83665136055644338798301571903, −21.97851291635424339552912611471, −21.52432801259975085556905097753, −20.19491338597048191505598514842, −19.12551127669975289278415604271, −17.82429036369665003468012319266, −16.96601129951676516863107148262, −15.464101513763045784735562084057, −14.7991172928609846143461459047, −13.93773049403272054163557909620, −12.19315107487828230070517036265, −11.10303539264208622935530786202, −9.80346952939546071640640168222, −9.24024456537259786198366951775, −7.629349251953432754229979705336, −6.01365775522590756503253676298, −4.98675431556220531314628494293, −3.3481115506731789474352705019, −2.20039355171017559727598354900, 0.807856555597561525302289828605, 1.905382847057236908395719947980, 3.79832972434805784896371468843, 5.43220906661786763596105927477, 6.69994082923590471914985291855, 7.84039378193290055091348382719, 8.98846909044616312708381830938, 10.226928777621463821234490134153, 11.90289147693870315268395243968, 12.63984295252982853316667455394, 14.10968213234678391799588317023, 14.22188815614844872150118513177, 16.61632844406319948098748541251, 17.021057405354984436718146420362, 18.21093528506076990133935840176, 19.45045167095352142098869959374, 20.22872024299341475526457137258, 21.25090686543013242182172130589, 22.65502079269768614028872596227, 23.82109922328982328077464847329, 24.5265298371172112092713614676, 25.31692362425016415445697464695, 26.50114433362220558707936919448, 27.63454186180702629292146552862, 28.96674093706742814127006903187

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