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Properties

Label 1-1312-1312.173-r0-0-0
Degree $1$
Conductor $1312$
Sign $0.799 - 0.600i$
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  + (0.707 − 0.707i)3-s + (−0.707 + 0.707i)5-s + 7-s − i·9-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s − i·15-s − i·17-s + (0.707 − 0.707i)19-s + (0.707 − 0.707i)21-s + i·23-s − i·25-s + (−0.707 − 0.707i)27-s + (0.707 − 0.707i)29-s + 31-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.707 + 0.707i)5-s + 7-s − i·9-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s − i·15-s − i·17-s + (0.707 − 0.707i)19-s + (0.707 − 0.707i)21-s + i·23-s − i·25-s + (−0.707 − 0.707i)27-s + (0.707 − 0.707i)29-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1312\)    =    \(2^{5} \cdot 41\)
Sign: $0.799 - 0.600i$
Analytic conductor: \(6.09290\)
Root analytic conductor: \(6.09290\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1312} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1312,\ (0:\ ),\ 0.799 - 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.759855150 - 0.5874750930i\)
\(L(\frac12)\) \(\approx\) \(1.759855150 - 0.5874750930i\)
\(L(1)\) \(\approx\) \(1.272539145 - 0.2219327028i\)
\(L(1)\) \(\approx\) \(1.272539145 - 0.2219327028i\)

Euler product

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

−20.755106809989591457852000832380, −20.36926538872102624099021755431, −19.90896697287667768144380603421, −18.79826882324335558576644107568, −18.045634539428271322807430189961, −17.1236206051991639605713202757, −16.26011882314194736934397150334, −15.66729286686267907809340088004, −14.94191127399853884424232985024, −14.33624248458931360634155669738, −13.435001390713271956201735419020, −12.45815736366491241001788639751, −11.822352874338765890665418261729, −10.817042212278007572248274020687, −10.05617427917534902639518511747, −9.300694156434522147364830981611, −8.190892069412417738838621664205, −7.99142050373134975596353815340, −7.1328310687400780989270245745, −5.35365654770494729507352856371, −4.85100188290383658029987483891, −4.31108578713328293912723351099, −3.07940670842250443181723979322, −2.32473789886588843825541748955, −0.99901750325768114571432940256, 0.83378869009088446426210329734, 2.08414542211254522955802425840, 2.78602202858461607918322410426, 3.74309685510217017592105672807, 4.648510806449237118824853509830, 5.85064140801740558443612175117, 6.806755488825498290579107694355, 7.67324153268917623660224198889, 8.00440948822886961519457253863, 8.88936285427436707518592186086, 9.929934866409598032848860479201, 10.988893508652504341093926115355, 11.60065252322349337012034907345, 12.27709410725951899118675538651, 13.3878076086130527686337277997, 14.01713797438675025490149737526, 14.673509013759347677934981620350, 15.35264408303074580956910189607, 16.0829724615176940371637117068, 17.54632611448115762563993745938, 17.74668993254175326636661729955, 18.92037704720934484140579522741, 19.1942389492493084540781744999, 19.89714723380173909927001886117, 20.92874119769657927159321710185

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