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Properties

Label 1-1312-1312.235-r0-0-0
Degree $1$
Conductor $1312$
Sign $-0.255 + 0.966i$
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  + 3-s + (−0.156 + 0.987i)5-s + (0.453 + 0.891i)7-s + 9-s + (−0.587 + 0.809i)11-s + (0.309 + 0.951i)13-s + (−0.156 + 0.987i)15-s + (−0.987 + 0.156i)17-s + (0.951 + 0.309i)19-s + (0.453 + 0.891i)21-s + (0.951 − 0.309i)23-s + (−0.951 − 0.309i)25-s + 27-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + ⋯
L(s)  = 1  + 3-s + (−0.156 + 0.987i)5-s + (0.453 + 0.891i)7-s + 9-s + (−0.587 + 0.809i)11-s + (0.309 + 0.951i)13-s + (−0.156 + 0.987i)15-s + (−0.987 + 0.156i)17-s + (0.951 + 0.309i)19-s + (0.453 + 0.891i)21-s + (0.951 − 0.309i)23-s + (−0.951 − 0.309i)25-s + 27-s + (0.809 − 0.587i)29-s + (−0.809 − 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1312\)    =    \(2^{5} \cdot 41\)
Sign: $-0.255 + 0.966i$
Analytic conductor: \(6.09290\)
Root analytic conductor: \(6.09290\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1312} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1312,\ (0:\ ),\ -0.255 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.358162066 + 1.763675941i\)
\(L(\frac12)\) \(\approx\) \(1.358162066 + 1.763675941i\)
\(L(1)\) \(\approx\) \(1.371534264 + 0.6221466013i\)
\(L(1)\) \(\approx\) \(1.371534264 + 0.6221466013i\)

Euler product

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

−20.59724830402590646037484926663, −19.93802020268264571903759240380, −19.6813603661749703220093530937, −18.38735049455283540173342017537, −17.83169480216092872624152088221, −16.82966376694831546734251135973, −15.98278206600435963316469435281, −15.544204870577007432085185394933, −14.50798399203718934300620113057, −13.62071271328270831004199278731, −13.25244550789000049829641895425, −12.55202463641412828138770947736, −11.239841188076430402511060468536, −10.67354327215511584450800691189, −9.57507861874898642641224299009, −8.89207957844697915832846661738, −8.057495301790660963590528768697, −7.67136594067767300488874413847, −6.578146067801579734515703046166, −5.16757808466157342292999250660, −4.70043222165985654481603567577, −3.56055482516628988822559821525, −2.93465602074290085058340558607, −1.548272914269236058573799687369, −0.771716539482614514905980202465, 1.707885977480670370892645018998, 2.36245126842738533802379094914, 3.09596158281637242995915802770, 4.16272129068463698421154624628, 4.96663452309388325634438189374, 6.26861812997802623552211765389, 7.04707508526352559358801265458, 7.78850906398785028327898659793, 8.60504489087955244235445409561, 9.41112245789509714370329125695, 10.12364172514051090888408006517, 11.150637149254662485832321005326, 11.80080214288965844142077562654, 12.844604947083618613495215589947, 13.62082521213481378976562008579, 14.41575349836362897406333543951, 15.093586376793102136350979469082, 15.45706429302637623267727055009, 16.382535356307189683107626053812, 17.77046713726336197970839340014, 18.33332109369009100162095285409, 18.7868921844307879463791099199, 19.62838404392740331408143106996, 20.42166078019409921666291529306, 21.20653849431992962848810714184

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