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Properties

Label 1-115-115.49-r0-0-0
Degree $1$
Conductor $115$
Sign $0.969 + 0.246i$
Analytic cond. $0.534057$
Root an. cond. $0.534057$
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  + (0.959 − 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (0.841 + 0.540i)6-s + (−0.415 + 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.959 + 0.281i)12-s + (−0.415 − 0.909i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)16-s + (−0.841 − 0.540i)17-s + (0.142 + 0.989i)18-s + (0.841 − 0.540i)19-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (0.841 + 0.540i)6-s + (−0.415 + 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.142 + 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.959 + 0.281i)12-s + (−0.415 − 0.909i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)16-s + (−0.841 − 0.540i)17-s + (0.142 + 0.989i)18-s + (0.841 − 0.540i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.969 + 0.246i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ 0.969 + 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.980150237 + 0.2476323802i\)
\(L(\frac12)\) \(\approx\) \(1.980150237 + 0.2476323802i\)
\(L(1)\) \(\approx\) \(1.879729000 + 0.1343703257i\)
\(L(1)\) \(\approx\) \(1.879729000 + 0.1343703257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.959 - 0.281i)T \)
3 \( 1 + (0.654 + 0.755i)T \)
7 \( 1 + (-0.415 + 0.909i)T \)
11 \( 1 + (-0.959 - 0.281i)T \)
13 \( 1 + (-0.415 - 0.909i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + (0.142 - 0.989i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.415 + 0.909i)T \)
59 \( 1 + (0.415 + 0.909i)T \)
61 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.841 + 0.540i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−29.2670361071812815401281811771, −28.956429731968224742275945014744, −26.56390229300427548942878122308, −26.14801088952366956127746741283, −25.0126398931165506979212530850, −23.953622901758684871155821804054, −23.45593164739063481513568088814, −22.25026725975938740918475438710, −20.90622788473146617348243915781, −20.161035858499076828039354763163, −19.16632207170599664623894208600, −17.736829428775919725948679965543, −16.54652541354359984565649345759, −15.37346750480404634772296694547, −14.240983684711627295276344514700, −13.43422453145973058679058373653, −12.66543743444954192705357454807, −11.42170348018392744789245676181, −9.87752371083076807599229981533, −8.1295968577824542104385851029, −7.21277015282741524914003077394, −6.27969777405129912993644233771, −4.5486136665772323458448269376, −3.29241165471799117319935083690, −1.96944304207132246694955544877, 2.5051670087058407109191465237, 3.16572550590239944408378936895, 4.8175236797638161429214700718, 5.63558903640417374502544075008, 7.39712143373924749022760878490, 8.91131455927985421757458722933, 10.121053723959437111735472358672, 11.131301072492232431388802193946, 12.54771153456977667013310046110, 13.487269077706016496086852286149, 14.61283835315357550598661075435, 15.70936990133606158454621069233, 16.0047065885619516310244852140, 18.107504215782457107239676770756, 19.46814217079188793974706268123, 20.212144249635677315546963012669, 21.29491207917708452567298973054, 22.033294278202229884449780623030, 22.84806464631969331671052606399, 24.360004055413994232592297018166, 25.12555195771801944015350402945, 26.12463048856187060227107149614, 27.37382187182488521467838632607, 28.519116681288543039100196428210, 29.286527894561011903270448287

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