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Properties

Label 2-3520-3520.1869-c0-0-2
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  + (−0.773 + 0.634i)2-s + (0.195 − 0.980i)4-s + (0.980 − 0.195i)5-s + (−0.360 − 0.871i)7-s + (0.471 + 0.881i)8-s + (0.382 − 0.923i)9-s + (−0.634 + 0.773i)10-s + (−0.555 + 0.831i)11-s + (0.192 + 0.0382i)13-s + (0.831 + 0.444i)14-s + (−0.923 − 0.382i)16-s + (−1.35 − 1.35i)17-s + (0.290 + 0.956i)18-s − i·20-s + (−0.0980 − 0.995i)22-s + ⋯
L(s)  = 1  + (−0.773 + 0.634i)2-s + (0.195 − 0.980i)4-s + (0.980 − 0.195i)5-s + (−0.360 − 0.871i)7-s + (0.471 + 0.881i)8-s + (0.382 − 0.923i)9-s + (−0.634 + 0.773i)10-s + (−0.555 + 0.831i)11-s + (0.192 + 0.0382i)13-s + (0.831 + 0.444i)14-s + (−0.923 − 0.382i)16-s + (−1.35 − 1.35i)17-s + (0.290 + 0.956i)18-s − i·20-s + (−0.0980 − 0.995i)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: $\chi_{3520} (1869, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ 0.471 + 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8494535907\)
\(L(\frac12)\) \(\approx\) \(0.8494535907\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.773 - 0.634i)T \)
5 \( 1 + (-0.980 + 0.195i)T \)
11 \( 1 + (0.555 - 0.831i)T \)
good3 \( 1 + (-0.382 + 0.923i)T^{2} \)
7 \( 1 + (0.360 + 0.871i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.192 - 0.0382i)T + (0.923 + 0.382i)T^{2} \)
17 \( 1 + (1.35 + 1.35i)T + iT^{2} \)
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 \( 1 + 1.11iT - T^{2} \)
37 \( 1 + (0.923 - 0.382i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (-1.05 - 0.704i)T + (0.382 + 0.923i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.382 + 0.923i)T^{2} \)
59 \( 1 + (1.92 - 0.382i)T + (0.923 - 0.382i)T^{2} \)
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 \( 1 + (-0.591 + 1.42i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + (-0.0382 + 0.192i)T + (-0.923 - 0.382i)T^{2} \)
89 \( 1 + (0.360 - 0.149i)T + (0.707 - 0.707i)T^{2} \)
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

−8.847696697560756817211597769200, −7.69605999266441772468775202969, −7.14277128283720777991016582908, −6.52408615224073494412442359810, −5.93033605570175714585428473270, −4.85653648036626052637084766863, −4.30677885396161432303982480010, −2.79338729432875077642192109980, −1.83391171225280683355676948516, −0.62867874128252294877770809220, 1.54737190424904292669768149908, 2.32749388102332860331254551019, 2.95863352797488012956802000282, 4.09567361504035034512885223319, 5.16864555958324970888741135539, 6.00730232066942225520695971631, 6.64331854540511035461551983227, 7.57690346006144974529322509443, 8.553973073280423232481795415068, 8.750256449248010570111531622358

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