Properties

Label 2-3120-3120.1283-c0-0-0
Degree $2$
Conductor $3120$
Sign $-0.348 - 0.937i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
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  + (0.258 + 0.965i)2-s + (−0.5 + 0.866i)3-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (−0.965 − 0.258i)6-s + (1.36 + 0.366i)7-s + (−0.707 − 0.707i)8-s + (−0.499 − 0.866i)9-s + (0.866 + 0.500i)10-s + (0.258 + 0.965i)11-s i·12-s i·13-s + 1.41i·14-s + (0.258 + 0.965i)15-s + (0.500 − 0.866i)16-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.5 + 0.866i)3-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (−0.965 − 0.258i)6-s + (1.36 + 0.366i)7-s + (−0.707 − 0.707i)8-s + (−0.499 − 0.866i)9-s + (0.866 + 0.500i)10-s + (0.258 + 0.965i)11-s i·12-s i·13-s + 1.41i·14-s + (0.258 + 0.965i)15-s + (0.500 − 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.348 - 0.937i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ -0.348 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.451951227\)
\(L(\frac12)\) \(\approx\) \(1.451951227\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + iT \)
good7 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.866 + 0.5i)T^{2} \)
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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

−9.192447912122913617683603146004, −8.151050432056309050877312119373, −7.87540601754926808161247150075, −6.60794255342969947089736846039, −5.84988221845115677382913619003, −5.22323978210998485943813522267, −4.74032972564868080732922679726, −4.13975532319983302932636185282, −2.79098954376131619908999599629, −1.26445566648540390783227709204, 1.11332047790144720059780133828, 1.89512352713148096680430031183, 2.68848925415201388764206273139, 3.83320387706965072786074022327, 4.93763980718568091148210924710, 5.37527846858287344650146534124, 6.50743692509067266865668605188, 6.80942123362558161465928524880, 8.178367704021486533240498900324, 8.485063556415850417930026108071

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