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Properties

Label 2-3520-3520.3189-c0-0-1
Degree $2$
Conductor $3520$
Sign $0.471 - 0.881i$
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  + (0.956 + 0.290i)2-s + (0.831 + 0.555i)4-s + (0.555 + 0.831i)5-s + (−1.17 − 0.485i)7-s + (0.634 + 0.773i)8-s + (0.923 − 0.382i)9-s + (0.290 + 0.956i)10-s + (−0.195 + 0.980i)11-s + (0.523 − 0.783i)13-s + (−0.980 − 0.804i)14-s + (0.382 + 0.923i)16-s + (0.138 − 0.138i)17-s + (0.995 − 0.0980i)18-s + i·20-s + (−0.471 + 0.881i)22-s + ⋯
L(s)  = 1  + (0.956 + 0.290i)2-s + (0.831 + 0.555i)4-s + (0.555 + 0.831i)5-s + (−1.17 − 0.485i)7-s + (0.634 + 0.773i)8-s + (0.923 − 0.382i)9-s + (0.290 + 0.956i)10-s + (−0.195 + 0.980i)11-s + (0.523 − 0.783i)13-s + (−0.980 − 0.804i)14-s + (0.382 + 0.923i)16-s + (0.138 − 0.138i)17-s + (0.995 − 0.0980i)18-s + i·20-s + (−0.471 + 0.881i)22-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (3189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ 0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.496299472\)
\(L(\frac12)\) \(\approx\) \(2.496299472\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.956 - 0.290i)T \)
5 \( 1 + (-0.555 - 0.831i)T \)
11 \( 1 + (0.195 - 0.980i)T \)
good3 \( 1 + (-0.923 + 0.382i)T^{2} \)
7 \( 1 + (1.17 + 0.485i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.523 + 0.783i)T + (-0.382 - 0.923i)T^{2} \)
17 \( 1 + (-0.138 + 0.138i)T - iT^{2} \)
19 \( 1 + (-0.382 - 0.923i)T^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.923 - 0.382i)T^{2} \)
31 \( 1 - 0.390iT - T^{2} \)
37 \( 1 + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.569 - 0.113i)T + (0.923 + 0.382i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.923 + 0.382i)T^{2} \)
59 \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \)
61 \( 1 + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (1.76 - 0.732i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + (-0.783 - 0.523i)T + (0.382 + 0.923i)T^{2} \)
89 \( 1 + (-0.636 + 1.53i)T + (-0.707 - 0.707i)T^{2} \)
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

−8.934794031479518189972289920615, −7.56784464429706794551748022247, −7.31766273985522800389653542201, −6.48225333532315551793567004268, −6.11919859900033216697937674958, −5.11725170473535128195440168594, −4.17266554723012649979033474377, −3.41967252141852668737813992097, −2.78803160547022134867217308328, −1.62581337666336148600460098166, 1.19019187939287302868211299575, 2.19111908787114854624049286157, 3.14926451695474591863325534533, 4.04224441529118180757922706182, 4.73357792733452951571160949804, 5.75866526708016669628809710935, 6.06150703337411486208131235216, 6.85348858492639991824751743909, 7.78099746879312613566673258276, 8.858353719045511835810869414407

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