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Properties

Label 1-3520-3520.1339-r0-0-0
Degree $1$
Conductor $3520$
Sign $-0.699 - 0.714i$
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  + (−0.852 − 0.522i)3-s + (−0.156 − 0.987i)7-s + (0.453 + 0.891i)9-s + (−0.0784 − 0.996i)13-s + (0.951 − 0.309i)17-s + (0.522 − 0.852i)19-s + (−0.382 + 0.923i)21-s + (0.707 − 0.707i)23-s + (0.0784 − 0.996i)27-s + (0.233 − 0.972i)29-s + (0.309 − 0.951i)31-s + (−0.522 − 0.852i)37-s + (−0.453 + 0.891i)39-s + (0.156 − 0.987i)41-s + (−0.382 + 0.923i)43-s + ⋯
L(s)  = 1  + (−0.852 − 0.522i)3-s + (−0.156 − 0.987i)7-s + (0.453 + 0.891i)9-s + (−0.0784 − 0.996i)13-s + (0.951 − 0.309i)17-s + (0.522 − 0.852i)19-s + (−0.382 + 0.923i)21-s + (0.707 − 0.707i)23-s + (0.0784 − 0.996i)27-s + (0.233 − 0.972i)29-s + (0.309 − 0.951i)31-s + (−0.522 − 0.852i)37-s + (−0.453 + 0.891i)39-s + (0.156 − 0.987i)41-s + (−0.382 + 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.699 - 0.714i$
Analytic conductor: \(16.3468\)
Root analytic conductor: \(16.3468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3520,\ (0:\ ),\ -0.699 - 0.714i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5048513840 - 1.200730619i\)
\(L(\frac12)\) \(\approx\) \(0.5048513840 - 1.200730619i\)
\(L(1)\) \(\approx\) \(0.7846950632 - 0.4145451665i\)
\(L(1)\) \(\approx\) \(0.7846950632 - 0.4145451665i\)

Euler product

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

−18.710555582649069205420499601914, −18.46878866801955836227035886757, −17.51189828922139650363515276590, −16.871538476842368017704065294365, −16.23656913851518242986212927613, −15.72798452647205780365776937456, −14.91123716602338053138363255890, −14.37663312332522672817037685979, −13.41643654636314971016264821522, −12.36010352636884111542894413328, −12.08999513859583582882481218867, −11.469651901651416881151675356992, −10.5783663300243170522322454707, −9.9216728183819907100515345746, −9.247074705652132697363469098983, −8.62568332529032165429887432356, −7.59144517333141084589157718421, −6.664916715936290368718204466, −6.1352911927694304336896694863, −5.162408762354137479463325199142, −4.971001116012318705308930781217, −3.64350534168054801170012231663, −3.23029952588851825673927117961, −1.89142658729012550036587878820, −1.1187776334735876359642116414, 0.61516296653157004210396028460, 0.912093405796386497429141305354, 2.24532304729868741622397379780, 3.10615759677600379605366921821, 4.10061104166042992354799849232, 4.91943634262500782915913477892, 5.57386172255271996277391022898, 6.37200307163292442372804729667, 7.17323315015221808198566878871, 7.6206432310613996736017481161, 8.37178686895685809011787730079, 9.64021849529456849207077961104, 10.11981688049927091352206360045, 10.94756191545244494713443648716, 11.37337363896132076837480685214, 12.360275162557210337276539122773, 12.85161934564675749601771883952, 13.55607760460375156204620693676, 14.13802198649444196697422114443, 15.11194765266336300531345612464, 15.94648974653563843822955869664, 16.48737582353232584206038691440, 17.35140400749348399104826473808, 17.529811979732721977888669140947, 18.41683685053157077573191591103

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