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Properties

Label 2-3520-3520.1429-c0-0-3
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  + (0.881 − 0.471i)2-s + (0.555 − 0.831i)4-s + (−0.831 + 0.555i)5-s + (0.181 + 0.0750i)7-s + (0.0980 − 0.995i)8-s + (−0.923 + 0.382i)9-s + (−0.471 + 0.881i)10-s + (−0.980 − 0.195i)11-s + (−1.59 − 1.06i)13-s + (0.195 − 0.0192i)14-s + (−0.382 − 0.923i)16-s + (−1.09 + 1.09i)17-s + (−0.634 + 0.773i)18-s + i·20-s + (−0.956 + 0.290i)22-s + ⋯
L(s)  = 1  + (0.881 − 0.471i)2-s + (0.555 − 0.831i)4-s + (−0.831 + 0.555i)5-s + (0.181 + 0.0750i)7-s + (0.0980 − 0.995i)8-s + (−0.923 + 0.382i)9-s + (−0.471 + 0.881i)10-s + (−0.980 − 0.195i)11-s + (−1.59 − 1.06i)13-s + (0.195 − 0.0192i)14-s + (−0.382 − 0.923i)16-s + (−1.09 + 1.09i)17-s + (−0.634 + 0.773i)18-s + i·20-s + (−0.956 + 0.290i)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: $\chi_{3520} (1429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ -0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1244826972\)
\(L(\frac12)\) \(\approx\) \(0.1244826972\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.881 + 0.471i)T \)
5 \( 1 + (0.831 - 0.555i)T \)
11 \( 1 + (0.980 + 0.195i)T \)
good3 \( 1 + (0.923 - 0.382i)T^{2} \)
7 \( 1 + (-0.181 - 0.0750i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (1.59 + 1.06i)T + (0.382 + 0.923i)T^{2} \)
17 \( 1 + (1.09 - 1.09i)T - iT^{2} \)
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 \( 1 - 1.96iT - T^{2} \)
37 \( 1 + (0.382 - 0.923i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.183 + 0.924i)T + (-0.923 - 0.382i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.923 - 0.382i)T^{2} \)
59 \( 1 + (1.38 - 0.923i)T + (0.382 - 0.923i)T^{2} \)
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 \( 1 + (-1.62 + 0.674i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + (-1.06 + 1.59i)T + (-0.382 - 0.923i)T^{2} \)
89 \( 1 + (0.425 - 1.02i)T + (-0.707 - 0.707i)T^{2} \)
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

−8.161694045425075958601670488669, −7.55897887542072222271946276697, −6.78481891327251148834811517303, −5.88611958245674744148180973948, −5.09254870050523863522257725702, −4.61317638115082202630022216010, −3.39933682856084030070730888112, −2.85052573071785043900093751090, −2.09212806490333753818362041755, −0.04670061922637619141555486746, 2.27117417559889315359832171227, 2.85591340416006339272439922765, 4.05349328202939786499044983193, 4.73172070375316493598389576646, 5.12686713822187074590277464067, 6.16678844136835223998087607449, 7.00714467231458309658237548816, 7.65331361426746790153822808928, 8.122498559885652489585900344490, 9.113032113042677103171640466872

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