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Properties

Label 2-3520-3520.2749-c0-0-2
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: $\chi_{3520} (2749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ -0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3941624566\)
\(L(\frac12)\) \(\approx\) \(0.3941624566\)
\(L(1)\) 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

−8.394937080736733704146986869976, −7.982962963474198730011335476891, −7.40470433588084471994568445978, −5.99775491165426905154350413904, −5.76610342101654991607990024448, −4.33087129860108165699495349941, −3.42443203570835542333720593642, −2.99667226979040955538448937907, −1.53391623261951728753801165881, −0.33688834519471922159447202798, 1.31497140453269917433308830572, 2.77345814708209492107785549609, 3.30393712392994904706269050308, 4.65598516887376904147243962153, 5.51716485830219567349842556721, 6.25185644833679080766130228122, 7.01112565415363711230806146624, 7.64763743803092363424263444141, 8.390756220503362632184621235737, 8.749746695719889914600748038428

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