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Properties

Label 2-3520-3520.549-c0-0-1
Degree $2$
Conductor $3520$
Sign $0.881 + 0.471i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
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  + (−0.995 + 0.0980i)2-s + (0.980 − 0.195i)4-s + (−0.195 + 0.980i)5-s + (−0.732 + 1.76i)7-s + (−0.956 + 0.290i)8-s + (−0.382 − 0.923i)9-s + (0.0980 − 0.995i)10-s + (0.831 − 0.555i)11-s + (−0.301 − 1.51i)13-s + (0.555 − 1.83i)14-s + (0.923 − 0.382i)16-s + (1.24 − 1.24i)17-s + (0.471 + 0.881i)18-s + i·20-s + (−0.773 + 0.634i)22-s + ⋯
L(s)  = 1  + (−0.995 + 0.0980i)2-s + (0.980 − 0.195i)4-s + (−0.195 + 0.980i)5-s + (−0.732 + 1.76i)7-s + (−0.956 + 0.290i)8-s + (−0.382 − 0.923i)9-s + (0.0980 − 0.995i)10-s + (0.831 − 0.555i)11-s + (−0.301 − 1.51i)13-s + (0.555 − 1.83i)14-s + (0.923 − 0.382i)16-s + (1.24 − 1.24i)17-s + (0.471 + 0.881i)18-s + i·20-s + (−0.773 + 0.634i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ 0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6253338646\)
\(L(\frac12)\) \(\approx\) \(0.6253338646\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.995 - 0.0980i)T \)
5 \( 1 + (0.195 - 0.980i)T \)
11 \( 1 + (-0.831 + 0.555i)T \)
good3 \( 1 + (0.382 + 0.923i)T^{2} \)
7 \( 1 + (0.732 - 1.76i)T + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.301 + 1.51i)T + (-0.923 + 0.382i)T^{2} \)
17 \( 1 + (-1.24 + 1.24i)T - iT^{2} \)
19 \( 1 + (-0.923 + 0.382i)T^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + 1.66iT - T^{2} \)
37 \( 1 + (-0.923 - 0.382i)T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + (0.108 + 0.162i)T + (-0.382 + 0.923i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.382 + 0.923i)T^{2} \)
59 \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \)
61 \( 1 + (0.382 + 0.923i)T^{2} \)
67 \( 1 + (0.382 + 0.923i)T^{2} \)
71 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.761 + 1.83i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + (-1.51 + 0.301i)T + (0.923 - 0.382i)T^{2} \)
89 \( 1 + (-1.81 - 0.750i)T + (0.707 + 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.783975848935380425135129173643, −8.013724441372410498951453153757, −7.33495838491745407547708316304, −6.37842707576226829060625439792, −5.95219333619361022647840703107, −5.43469418361824282571823958287, −3.38127183178160074157701052570, −3.09256624814342730961089310044, −2.34316590287533258954250781273, −0.56749717942169653541890048954, 1.18018323233271709546817807736, 1.81537204759428718488058196398, 3.40556433546776313258021919152, 4.05944494320363260603833371805, 4.88283564545243785954627608559, 6.11267548494346838251089510303, 6.85450236215001206914728531523, 7.43160715366936794741209570271, 8.083924774117628803538439506402, 8.850330074264575021744444381935

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