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Properties

Label 1-5160-5160.1253-r0-0-0
Degree $1$
Conductor $5160$
Sign $0.716 - 0.697i$
Analytic cond. $23.9629$
Root an. cond. $23.9629$
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  + (0.866 + 0.5i)7-s + 11-s + (0.866 + 0.5i)13-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.866 − 0.5i)37-s − 41-s + i·47-s + (0.5 + 0.866i)49-s + (0.866 − 0.5i)53-s − 59-s + (0.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + 11-s + (0.866 + 0.5i)13-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s + (0.866 − 0.5i)37-s − 41-s + i·47-s + (0.5 + 0.866i)49-s + (0.866 − 0.5i)53-s − 59-s + (0.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5160\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 43\)
Sign: $0.716 - 0.697i$
Analytic conductor: \(23.9629\)
Root analytic conductor: \(23.9629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5160} (1253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5160,\ (0:\ ),\ 0.716 - 0.697i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.863470637 - 0.7575706746i\)
\(L(\frac12)\) \(\approx\) \(1.863470637 - 0.7575706746i\)
\(L(1)\) \(\approx\) \(1.230431759 - 0.07243586433i\)
\(L(1)\) \(\approx\) \(1.230431759 - 0.07243586433i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
43 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 - T \)
47 \( 1 + iT \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 - T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−18.10402470261655256602331883674, −17.42073805680868202124984985445, −16.76675030563658965194337855083, −16.22805675423235829240775521325, −15.291775418774527615798422593017, −14.725895452372001780473467899771, −14.12422275183999311726521707612, −13.52499198781055362014729972868, −12.72665647266150326871850517169, −12.003825841387273779246650796496, −11.36526982275151496200686422901, −10.58179694302623735654433660672, −10.27547959669307264139472631718, −9.14933465999379662436293200272, −8.37056539299386685564403337034, −8.18940579977044854937671622787, −7.00160823253924008584459207754, −6.519886173207332961057798133858, −5.71209554120095364652117819506, −4.8866033174282532791309212254, −3.95205879827457090206780411070, −3.76122997699819414950239551803, −2.46786415036735679802591445664, −1.57668601083826184961907222427, −1.0431569461669923147484171646, 0.56457543855206501714475476184, 1.73461443417435994599066324594, 2.13181387967417248148022946449, 3.20890820010308816312784997667, 4.26678641904771004174850986286, 4.47894466962777941189925736334, 5.59231346563639109849561250852, 6.261440574796637011847475388718, 6.855929942834962720825036550, 7.81832042500875900384961584530, 8.44125345228486579414493551195, 9.168245609558967470107202541509, 9.5495964831650884105817653060, 10.749433685627586835335986303917, 11.35613174804297161685825342918, 11.67054455364103930975859133441, 12.475289707450334666947957200976, 13.49930378213584288526219392515, 13.798915839084567250837178091903, 14.66645853916677759240224543713, 15.2322693968138875032899747310, 15.85975490913104540303018153501, 16.59667759618472216003696941352, 17.424217047448812568177155141606, 17.81383280475785655036469440272

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