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Properties

Label 1-3520-3520.1149-r0-0-0
Degree $1$
Conductor $3520$
Sign $0.0131 + 0.999i$
Analytic cond. $16.3468$
Root an. cond. $16.3468$
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  + (−0.0784 + 0.996i)3-s + (0.453 + 0.891i)7-s + (−0.987 − 0.156i)9-s + (−0.233 − 0.972i)13-s + (−0.587 + 0.809i)17-s + (−0.996 − 0.0784i)19-s + (−0.923 + 0.382i)21-s + (0.707 + 0.707i)23-s + (0.233 − 0.972i)27-s + (0.649 − 0.760i)29-s + (0.809 − 0.587i)31-s + (0.996 − 0.0784i)37-s + (0.987 − 0.156i)39-s + (0.453 − 0.891i)41-s + (0.923 − 0.382i)43-s + ⋯
L(s)  = 1  + (−0.0784 + 0.996i)3-s + (0.453 + 0.891i)7-s + (−0.987 − 0.156i)9-s + (−0.233 − 0.972i)13-s + (−0.587 + 0.809i)17-s + (−0.996 − 0.0784i)19-s + (−0.923 + 0.382i)21-s + (0.707 + 0.707i)23-s + (0.233 − 0.972i)27-s + (0.649 − 0.760i)29-s + (0.809 − 0.587i)31-s + (0.996 − 0.0784i)37-s + (0.987 − 0.156i)39-s + (0.453 − 0.891i)41-s + (0.923 − 0.382i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0131 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0131 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.0131 + 0.999i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ 0.0131 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.115061794 + 1.129816741i\)
\(L(\frac12)\) \(\approx\) \(1.115061794 + 1.129816741i\)
\(L(1)\) \(\approx\) \(0.9683383815 + 0.4269458418i\)
\(L(1)\) \(\approx\) \(0.9683383815 + 0.4269458418i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.0784 + 0.996i)T \)
7 \( 1 + (0.453 + 0.891i)T \)
13 \( 1 + (-0.233 - 0.972i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.996 - 0.0784i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.649 - 0.760i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.996 - 0.0784i)T \)
41 \( 1 + (0.453 - 0.891i)T \)
43 \( 1 + (0.923 - 0.382i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.522 - 0.852i)T \)
59 \( 1 + (0.996 - 0.0784i)T \)
61 \( 1 + (0.522 + 0.852i)T \)
67 \( 1 + (0.923 + 0.382i)T \)
71 \( 1 + (-0.987 + 0.156i)T \)
73 \( 1 + (-0.891 + 0.453i)T \)
79 \( 1 + (0.587 + 0.809i)T \)
83 \( 1 + (-0.852 + 0.522i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−18.5718435507620020302600741272, −17.84269357467840844038777775963, −17.28642008463564186927946771312, −16.66186319044825139579243180886, −16.02454332798721847424252966066, −14.81564188967492927451929818703, −14.30707112740632488534878536762, −13.749701184075704975987586084591, −13.03247539044449593127412550650, −12.39937799059283359926600274096, −11.558266079273804016158319634651, −11.04434504008326602691454011545, −10.34425050738940283021514167875, −9.245780454156788723882775403603, −8.60016001123921809356828759434, −7.868674391288950770774759199765, −6.99157565858696625956511033506, −6.75161609238268484281870725813, −5.8457137929073544938965338104, −4.63341550629323044093949788277, −4.394700316801307210465942972731, −2.997094139905684504730455389, −2.3331175290976552409723782447, −1.39741352821707095085828607265, −0.62894069327265499872132764393, 0.824467601197573682144763563192, 2.34936138438730378892273043008, 2.64849509051088288217223407163, 3.90445938124727768740835869240, 4.374596579294348066655960915930, 5.41664244379277436576792686844, 5.729392581779130441280597956699, 6.64940634929888190688945891381, 7.846039184135940610652010410685, 8.48248602236066340487713190816, 8.99872936403497477231858349227, 9.87773401767310858584974064877, 10.50627940777732498366382429538, 11.19180896871258127561928978908, 11.79953861217518534350349919353, 12.65483339087021213326215233028, 13.32318446713308460281609375301, 14.360278854853934098062063426830, 15.01019336398094237385832170883, 15.39519488797885935289246911961, 15.912007877353453278448627125086, 17.03837588639546310738731796212, 17.41341465424279160747535604131, 17.980624714678514125231382497114, 19.16785127072169536241192138269

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