Properties

Label 2-3520-3520.1429-c0-0-1
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  + (−0.471 − 0.881i)2-s + (−0.555 + 0.831i)4-s + (0.831 − 0.555i)5-s + (1.83 + 0.761i)7-s + (0.995 + 0.0980i)8-s + (−0.923 + 0.382i)9-s + (−0.881 − 0.471i)10-s + (0.980 + 0.195i)11-s + (0.482 + 0.322i)13-s + (−0.195 − 1.98i)14-s + (−0.382 − 0.923i)16-s + (−0.897 + 0.897i)17-s + (0.773 + 0.634i)18-s + i·20-s + (−0.290 − 0.956i)22-s + ⋯
L(s)  = 1  + (−0.471 − 0.881i)2-s + (−0.555 + 0.831i)4-s + (0.831 − 0.555i)5-s + (1.83 + 0.761i)7-s + (0.995 + 0.0980i)8-s + (−0.923 + 0.382i)9-s + (−0.881 − 0.471i)10-s + (0.980 + 0.195i)11-s + (0.482 + 0.322i)13-s + (−0.195 − 1.98i)14-s + (−0.382 − 0.923i)16-s + (−0.897 + 0.897i)17-s + (0.773 + 0.634i)18-s + i·20-s + (−0.290 − 0.956i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.365719218\)
\(L(\frac12)\) \(\approx\) \(1.365719218\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.471 + 0.881i)T \)
5 \( 1 + (-0.831 + 0.555i)T \)
11 \( 1 + (-0.980 - 0.195i)T \)
good3 \( 1 + (0.923 - 0.382i)T^{2} \)
7 \( 1 + (-1.83 - 0.761i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.482 - 0.322i)T + (0.382 + 0.923i)T^{2} \)
17 \( 1 + (0.897 - 0.897i)T - iT^{2} \)
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 \( 1 + (0.344 - 1.72i)T + (-0.923 - 0.382i)T^{2} \)
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 \( 1 + (0.871 - 0.360i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + (0.322 - 0.482i)T + (-0.382 - 0.923i)T^{2} \)
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

−8.892281124466112631488000183966, −8.255446977684485836862223127372, −7.66207017966694143145151818505, −6.22801333790969524241928518759, −5.65298152779149544711191150293, −4.57432996452974775120744090910, −4.30743779237906325060057353006, −2.75142996869642749304531677468, −1.89821120383192423275632145195, −1.45766682760243129413332106012, 1.11093024978031131171729688287, 1.96872960749537090044614560107, 3.39240488938107693325450767183, 4.50980797917004017658775576286, 5.18937859100794343839062360052, 5.88434974756939255096402448989, 6.76840918577435636245774483739, 7.17590767310423170246527197389, 8.140509142809544914455671195157, 8.826087663677672526484568187323

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