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Properties

Label 2-3520-3520.1429-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  + (0.471 + 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

\(L(\frac{1}{2})\) \(\approx\) \(1.069056297\)
\(L(\frac12)\) \(\approx\) \(1.069056297\)
\(L(1)\) 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

−8.830744556608251956794940345425, −7.65964018874792018335304706246, −7.19059046148112833721060262140, −6.16793095799669416351147344538, −5.97856111480105946958292561744, −5.07063681808115752823768354596, −4.13158536336522621022861225409, −3.30342069549378816326363936332, −2.51946919975872152474476988165, −0.53164600090731880818728385414, 1.44595954333962190877622267943, 2.67352398509476581253261323021, 3.14832622465120304826671478722, 3.75673811920275665784792823433, 5.13974428501560626547589609746, 5.97156832538684545107554429742, 6.25557895769283110159350413058, 6.88150045991212660907528797231, 8.476233396057770709394475979396, 9.203066472201386988355844437124

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