Properties

Label 1-116-116.99-r0-0-0
Degree $1$
Conductor $116$
Sign $-0.981 - 0.189i$
Analytic cond. $0.538701$
Root an. cond. $0.538701$
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  i·3-s − 5-s − 7-s − 9-s i·11-s − 13-s + i·15-s + i·17-s i·19-s + i·21-s − 23-s + 25-s + i·27-s i·31-s − 33-s + ⋯
L(s)  = 1  i·3-s − 5-s − 7-s − 9-s i·11-s − 13-s + i·15-s + i·17-s i·19-s + i·21-s − 23-s + 25-s + i·27-s i·31-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $-0.981 - 0.189i$
Analytic conductor: \(0.538701\)
Root analytic conductor: \(0.538701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 116,\ (0:\ ),\ -0.981 - 0.189i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03733009779 - 0.3912410531i\)
\(L(\frac12)\) \(\approx\) \(0.03733009779 - 0.3912410531i\)
\(L(1)\) \(\approx\) \(0.5391112713 - 0.3042919672i\)
\(L(1)\) \(\approx\) \(0.5391112713 - 0.3042919672i\)

Euler product

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

−29.556113617516623894141016152245, −28.52095753225896007220757718666, −27.584213607499558887524707616854, −26.80492998762356080603862130668, −25.891893017225729278297468771661, −24.790527416825049408149010192253, −23.12613266161986415287110408250, −22.75230642086534898789815310603, −21.66408146334397881545216583011, −20.16754697113589318577854192605, −19.85337528992023898878098534267, −18.43489497682181113651312141054, −16.894133094382135384660853712788, −16.0701813867953237048504404120, −15.258601417960201156259534904783, −14.24600489399605487291621377095, −12.4947064476980806851363850591, −11.706261860876908796592832471973, −10.189151841839005927393591256, −9.53694293954025856428085568343, −8.05678606375511229368794689761, −6.77921202709610149935270001362, −5.07807663590806382556886208505, −4.005944547201673793635139517448, −2.84084847949748106858518580912, 0.35577537082832813059359961519, 2.55932305193514134932984533227, 3.83693375030246398657943032037, 5.76552553730440870922604223651, 6.91731204772854312514523580197, 7.91652383812084888996225059713, 9.052499128078792571058678567824, 10.75786821371137973335390480926, 11.94463216607519720318120543897, 12.7311753892512426175959059910, 13.82169363271607112607229622437, 15.13477001047866419986662386546, 16.30087174821338303288836270530, 17.31677024798406244971048448538, 18.76366779145127143528572761515, 19.41159586220681616452046285650, 20.0176049982896823717278034276, 21.89112199187841306323766723245, 22.74966161124073266197178575529, 23.951316273190174298772741533507, 24.32668463269994153183015937783, 25.82784782402170578870931769270, 26.55049256990118166865460511470, 27.90936275485432228407726252360, 28.90228116292172534744978175665

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