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Properties

Label 1-3751-3751.815-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.978 - 0.205i$
Analytic cond. $17.4195$
Root an. cond. $17.4195$
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  + (0.610 + 0.791i)2-s + (0.913 + 0.406i)3-s + (−0.254 + 0.967i)4-s + (−0.327 − 0.945i)5-s + (0.235 + 0.971i)6-s + (−0.398 − 0.917i)7-s + (−0.921 + 0.389i)8-s + (0.669 + 0.743i)9-s + (0.548 − 0.836i)10-s + (−0.625 + 0.780i)12-s + (0.449 − 0.893i)13-s + (0.483 − 0.875i)14-s + (0.0855 − 0.996i)15-s + (−0.870 − 0.491i)16-s + (0.345 + 0.938i)17-s + (−0.179 + 0.983i)18-s + ⋯
L(s)  = 1  + (0.610 + 0.791i)2-s + (0.913 + 0.406i)3-s + (−0.254 + 0.967i)4-s + (−0.327 − 0.945i)5-s + (0.235 + 0.971i)6-s + (−0.398 − 0.917i)7-s + (−0.921 + 0.389i)8-s + (0.669 + 0.743i)9-s + (0.548 − 0.836i)10-s + (−0.625 + 0.780i)12-s + (0.449 − 0.893i)13-s + (0.483 − 0.875i)14-s + (0.0855 − 0.996i)15-s + (−0.870 − 0.491i)16-s + (0.345 + 0.938i)17-s + (−0.179 + 0.983i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.978 - 0.205i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.978 - 0.205i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.508074784 - 0.2598424650i\)
\(L(\frac12)\) \(\approx\) \(2.508074784 - 0.2598424650i\)
\(L(1)\) \(\approx\) \(1.585846093 + 0.4234918516i\)
\(L(1)\) \(\approx\) \(1.585846093 + 0.4234918516i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.610 + 0.791i)T \)
3 \( 1 + (0.913 + 0.406i)T \)
5 \( 1 + (-0.327 - 0.945i)T \)
7 \( 1 + (-0.398 - 0.917i)T \)
13 \( 1 + (0.449 - 0.893i)T \)
17 \( 1 + (0.345 + 0.938i)T \)
19 \( 1 + (-0.969 - 0.244i)T \)
23 \( 1 + (0.993 - 0.113i)T \)
29 \( 1 + (0.696 - 0.717i)T \)
37 \( 1 + (-0.888 - 0.458i)T \)
41 \( 1 + (-0.991 + 0.132i)T \)
43 \( 1 + (0.820 - 0.572i)T \)
47 \( 1 + (0.610 - 0.791i)T \)
53 \( 1 + (0.953 - 0.299i)T \)
59 \( 1 + (-0.991 - 0.132i)T \)
61 \( 1 + (-0.959 + 0.281i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (-0.830 + 0.556i)T \)
73 \( 1 + (-0.217 - 0.976i)T \)
79 \( 1 + (0.483 - 0.875i)T \)
83 \( 1 + (0.00951 - 0.999i)T \)
89 \( 1 + (0.897 - 0.441i)T \)
97 \( 1 + (0.516 - 0.856i)T \)
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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

−18.788180150892074038892292280, −18.51092390624490427464008557887, −17.65651724704572842697421482154, −16.28796226844014230109034971499, −15.46252386318297080966693317792, −15.12138649405706629834142434019, −14.30509375008925014200401581641, −13.92459385604378005595240573026, −13.168370487917941830611641175985, −12.3085314771473867781184556778, −11.96544920970007719375441990087, −11.09046489691287134887378340467, −10.39563813652268648811031666922, −9.51708159028466523330883329094, −8.998759136516678115389230448717, −8.29912711906336578842886594790, −7.098449226915897067706400763333, −6.6454503779098448837739469435, −5.92327187864336463333078810009, −4.83766251483165039219993837111, −3.97331561280300375619697104782, −3.213813185694768737573600949634, −2.741711478798538080643111667958, −2.074612837537332224757102804371, −1.14666480079410905527075737567, 0.51731875360507449978047901975, 1.78092728757187075051285761398, 2.99785509497460961875991514108, 3.637930818114496188514913708556, 4.20851053293354553429069891818, 4.83841905731900468443191767366, 5.68157099921641134372456100301, 6.60537191785314849363951315450, 7.46231134778708359002494483564, 8.01399339236051908150450718414, 8.67791009054493017674039157167, 9.129293468677713669428596086201, 10.27989223519901373831360083722, 10.73702585991029623767812691666, 12.038769981811101126851354076640, 12.73295868157026831530714702145, 13.29990869059732634660875529634, 13.63359639316863401259039622668, 14.5961961693337634275984409926, 15.275185863673598084011557307717, 15.66780713797426339454605344160, 16.42686966189986902465602154260, 17.06870550290580340520622478394, 17.432907706925390900849587871185, 18.72634226866961249104522207323

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