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Properties

Label 1-1312-1312.1155-r0-0-0
Degree $1$
Conductor $1312$
Sign $0.999 + 0.0226i$
Analytic cond. $6.09290$
Root an. cond. $6.09290$
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 + (0.453 − 0.891i)5-s + (−0.156 + 0.987i)7-s − 9-s + (−0.309 + 0.951i)11-s + (0.587 − 0.809i)13-s + (0.891 + 0.453i)15-s + (−0.453 − 0.891i)17-s + (0.809 − 0.587i)19-s + (−0.987 − 0.156i)21-s + (0.587 − 0.809i)23-s + (−0.587 − 0.809i)25-s − i·27-s + (0.951 − 0.309i)29-s + (0.309 − 0.951i)31-s + ⋯
L(s)  = 1  − i·3-s + (0.453 − 0.891i)5-s + (−0.156 + 0.987i)7-s − 9-s + (−0.309 + 0.951i)11-s + (0.587 − 0.809i)13-s + (0.891 + 0.453i)15-s + (−0.453 − 0.891i)17-s + (0.809 − 0.587i)19-s + (−0.987 − 0.156i)21-s + (0.587 − 0.809i)23-s + (−0.587 − 0.809i)25-s − i·27-s + (0.951 − 0.309i)29-s + (0.309 − 0.951i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1312\)    =    \(2^{5} \cdot 41\)
Sign: $0.999 + 0.0226i$
Analytic conductor: \(6.09290\)
Root analytic conductor: \(6.09290\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1312} (1155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1312,\ (0:\ ),\ 0.999 + 0.0226i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.523778153 + 0.01727636007i\)
\(L(\frac12)\) \(\approx\) \(1.523778153 + 0.01727636007i\)
\(L(1)\) \(\approx\) \(1.106655765 + 0.1566763816i\)
\(L(1)\) \(\approx\) \(1.106655765 + 0.1566763816i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 - iT \)
5 \( 1 + (0.453 - 0.891i)T \)
7 \( 1 + (-0.156 + 0.987i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
17 \( 1 + (-0.453 - 0.891i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (0.951 - 0.309i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.891 - 0.453i)T \)
43 \( 1 + (-0.156 - 0.987i)T \)
47 \( 1 + (0.987 - 0.156i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.987 + 0.156i)T \)
61 \( 1 + (-0.156 + 0.987i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (-0.453 - 0.891i)T \)
73 \( 1 + T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (0.156 - 0.987i)T \)
97 \( 1 + (0.891 + 0.453i)T \)
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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

−21.051638484594639391334052999245, −19.976546280597905603476928678582, −19.26268940518659338053432177559, −18.7762785360608729143440234064, −17.94149749434716894026478420451, −17.34352642947735306050672777647, −16.556901421655512061481748173973, −15.65975918922475329018169666999, −14.46868493342840486205791870311, −13.767242011627806178852023644362, −13.63350475007908708678636567149, −12.59978729410447829193709781494, −11.52337211239320685720760583475, −10.92989298410708995520225020255, −10.25604594804516336978883889192, −9.116354644761683019340400966325, −8.20375635835193956530774781065, −7.413134093920024610675480468762, −6.563089388372589572893095584507, −6.20954735207772315627390217762, −5.08807931394357700343734341763, −3.57358797246878964385921509799, −3.12947320711772710425303012207, −1.83542965101357097436841196876, −1.08586480243833441381918479010, 0.68393650524304458916557189982, 2.285896259594069895613042921146, 2.88561058779083116158680447055, 4.19201589273890992850299612138, 5.02923669759800640751210463034, 5.44100328994588523964119635002, 6.40932710075712983053725175093, 7.71979072449669609718730260753, 8.7935682045187980684892718158, 9.09403020932494765898118106566, 9.94668656498467620323788302054, 10.66272100544479220163738753595, 11.78317726631601946122605274673, 12.33780345618668730247279733568, 13.30489898279664532932454106645, 14.00295523915103650721115467394, 15.265768099905375966514160929084, 15.51763867338232724379302515053, 16.20107532060987946039277536662, 17.15307943340867146366998103387, 17.809817478216366242477442038667, 18.497482095597351219566233383038, 19.77035101091529340416199149270, 20.43692980714140880768661553434, 20.84727873160830486343787545744

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