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Properties

Label 1-1312-1312.11-r0-0-0
Degree $1$
Conductor $1312$
Sign $-0.759 - 0.650i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1312\)    =    \(2^{5} \cdot 41\)
Sign: $-0.759 - 0.650i$
Analytic conductor: \(6.09290\)
Root analytic conductor: \(6.09290\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1312} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1312,\ (0:\ ),\ -0.759 - 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1531962550 - 0.4145460010i\)
\(L(\frac12)\) \(\approx\) \(0.1531962550 - 0.4145460010i\)
\(L(1)\) \(\approx\) \(0.6466512751 - 0.09126143458i\)
\(L(1)\) \(\approx\) \(0.6466512751 - 0.09126143458i\)

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

−21.44673950331927906472847639120, −20.65532067549097646823607281731, −19.55833082089598210470879107905, −18.97879389338956359920338152099, −18.22049075066933619524296592546, −17.11957587445414345919179538468, −16.82202772201808945396184503391, −16.14153192180165844893449771768, −15.216379609209437542608574919870, −14.58752995537576604675265334201, −13.09640893179200307171663272696, −12.70601685792378132967449176514, −11.91451823377796913146845869159, −11.54867429981058439145117897485, −10.27829181850265404705581099657, −9.437872975534758036222885910269, −8.98128024811207521679428284517, −7.756963845967973367490196345897, −6.85679404620711410550602056614, −6.01462307984807144803811723115, −5.26936863985522540318584194181, −4.50348853069197855810668577407, −3.68883736905495606066898792857, −2.070788962467666289557585496254, −1.32034223219668622452921359134, 0.22047015876303465851696976081, 1.28644381390424830658243501242, 2.85744593849027809220752712388, 3.618462411865698556934103731206, 4.492294467954025528482216519057, 5.72496326026981272956518171004, 6.23436263111427346653194297829, 7.22753407669079355685390406285, 7.58428641519960031125245837876, 9.03444708627668965227621364827, 10.07714709589209346052897464218, 10.66039338704637419856658049890, 11.128728867536167437044920825306, 12.05875788999461639070002751078, 12.9477185050984710741936257903, 13.61546182581513049585912116078, 14.72876264075568267347916730959, 15.19273402436480275209395529413, 16.43177211445250862288798714988, 16.77829042051454515457463662541, 17.511754871623883085700748803193, 18.48063233955989620315834508849, 18.96849148724180033462868032350, 19.76734405195556181472255740447, 20.70722494296672022003393939312

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