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Properties

Label 1-912-912.341-r0-0-0
Degree $1$
Conductor $912$
Sign $-0.923 + 0.382i$
Analytic cond. $4.23531$
Root an. cond. $4.23531$
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·5-s − 7-s − i·11-s + i·13-s − 17-s + 23-s − 25-s − i·29-s − 31-s + i·35-s − i·37-s − 41-s + i·43-s − 47-s + 49-s + ⋯
L(s)  = 1  − i·5-s − 7-s − i·11-s + i·13-s − 17-s + 23-s − 25-s − i·29-s − 31-s + i·35-s − i·37-s − 41-s + i·43-s − 47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.923 + 0.382i$
Analytic conductor: \(4.23531\)
Root analytic conductor: \(4.23531\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 912,\ (0:\ ),\ -0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03207301956 - 0.1612419578i\)
\(L(\frac12)\) \(\approx\) \(-0.03207301956 - 0.1612419578i\)
\(L(1)\) \(\approx\) \(0.7095003023 - 0.1692810020i\)
\(L(1)\) \(\approx\) \(0.7095003023 - 0.1692810020i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
13 \( 1 - iT \)
17 \( 1 \)
23 \( 1 \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - T \)
67 \( 1 \)
71 \( 1 \)
73 \( 1 \)
79 \( 1 \)
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

−22.29868459290751210205806379176, −21.94431244813421058738096923276, −20.58173976787254341452369339412, −19.97009745670157217407750532180, −19.20901442118531247834671442913, −18.3466884258489133193644946558, −17.74805741098329492868993304312, −16.87123686627975838839683913841, −15.771611023247620873375803945719, −15.1912986973107134128445716334, −14.56418135755229362564844576613, −13.318863934915396155888162550286, −12.897712848460233783206704993014, −11.863397443621354831916757892848, −10.83478251319162140422235160960, −10.22340020386248252123500516087, −9.460546618153084305625460987342, −8.438608126799625471367392065394, −7.09822680684534378074947238708, −6.91841589577896261083939066754, −5.80687138966914872353376425587, −4.75771982780033504743481753401, −3.49721000154966881427028982218, −2.911593629136343502652164021340, −1.80958326230970197282563663903, 0.07016082138017918572840162398, 1.39165835829592452340350173144, 2.63114434660372000307565183616, 3.75010033478233882491700542085, 4.55256003440911667478757403890, 5.653268544756029790158701451200, 6.41144160460394069443955539969, 7.355825374698261904405231148253, 8.62970553829063883852367762549, 9.04608685066979257100381281096, 9.84907302659565702719493583201, 11.06293250163664640835781476807, 11.72988436094166876497594625760, 12.8213360444485468200922259853, 13.26585367129843581090401058192, 14.05665846717532578854385184350, 15.2624392465460370784282975587, 16.11382346996571897733934954191, 16.56288143847648628344033323355, 17.27943467930186949822004532291, 18.425607619717658223161143287263, 19.29776223183495904705367613921, 19.71560661177568783352329681266, 20.71810596851773338123238410817, 21.476909524479673616630276075191

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