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Properties

Label 2-3520-3520.109-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  + (0.0980 − 0.995i)2-s + (−0.980 − 0.195i)4-s + (0.195 + 0.980i)5-s + (−0.222 − 0.536i)7-s + (−0.290 + 0.956i)8-s + (−0.382 + 0.923i)9-s + (0.995 − 0.0980i)10-s + (−0.831 − 0.555i)11-s + (0.247 − 1.24i)13-s + (−0.555 + 0.168i)14-s + (0.923 + 0.382i)16-s + (−0.666 − 0.666i)17-s + (0.881 + 0.471i)18-s − i·20-s + (−0.634 + 0.773i)22-s + ⋯
L(s)  = 1  + (0.0980 − 0.995i)2-s + (−0.980 − 0.195i)4-s + (0.195 + 0.980i)5-s + (−0.222 − 0.536i)7-s + (−0.290 + 0.956i)8-s + (−0.382 + 0.923i)9-s + (0.995 − 0.0980i)10-s + (−0.831 − 0.555i)11-s + (0.247 − 1.24i)13-s + (−0.555 + 0.168i)14-s + (0.923 + 0.382i)16-s + (−0.666 − 0.666i)17-s + (0.881 + 0.471i)18-s − i·20-s + (−0.634 + 0.773i)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: $\chi_{3520} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ -0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7263149118\)
\(L(\frac12)\) \(\approx\) \(0.7263149118\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0980 + 0.995i)T \)
5 \( 1 + (-0.195 - 0.980i)T \)
11 \( 1 + (0.831 + 0.555i)T \)
good3 \( 1 + (0.382 - 0.923i)T^{2} \)
7 \( 1 + (0.222 + 0.536i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.247 + 1.24i)T + (-0.923 - 0.382i)T^{2} \)
17 \( 1 + (0.666 + 0.666i)T + iT^{2} \)
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 \( 1 + (-1.10 + 1.65i)T + (-0.382 - 0.923i)T^{2} \)
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 \( 1 + (-0.0750 + 0.181i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + (1.24 + 0.247i)T + (0.923 + 0.382i)T^{2} \)
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

−8.422386399731663894871237102354, −7.81928214315652968073840550211, −7.12170215968501143685562587492, −5.81906849337050645252095862069, −5.51893932724620066908619387726, −4.43061705315332437330846254350, −3.48640240862141362826807337976, −2.74517173323142675600519245611, −2.18133419076696237632011671849, −0.41617375716728551864830525526, 1.40018568343111515626966917206, 2.78014653332554751652298743422, 4.01642461370335490989057931794, 4.57314978184906069371384022793, 5.42590851047640746798341928321, 6.10334915664328358498261962413, 6.69709229367137536170113803992, 7.55964312722259626100066494166, 8.558889065587913242657413944978, 8.813815142037929944027030944964

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