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Properties

Label 2-3120-3120.2963-c0-0-6
Degree $2$
Conductor $3120$
Sign $-0.160 + 0.987i$
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  − i·2-s + 3-s − 4-s + 5-s − i·6-s + i·8-s + 9-s − i·10-s + (−1 − i)11-s − 12-s − i·13-s + 15-s + 16-s − i·18-s − 20-s + ⋯
L(s)  = 1  − i·2-s + 3-s − 4-s + 5-s − i·6-s + i·8-s + 9-s − i·10-s + (−1 − i)11-s − 12-s − i·13-s + 15-s + 16-s − i·18-s − 20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.160 + 0.987i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (2963, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ -0.160 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.844446538\)
\(L(\frac12)\) \(\approx\) \(1.844446538\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.662619783091493859067188349189, −8.279278556162001924311673605817, −7.49544336034108061658455259527, −6.23594173499046276770766549568, −5.41349323283068553572615381465, −4.73741575966261569694415275689, −3.50830179108020617865847865999, −2.91449158227643011550084916224, −2.24244777211437787107058008926, −1.10242295363192986271165330554, 1.65175507396444904373510744622, 2.51361146193342594907370237039, 3.67382990029089376153461557608, 4.66987706946254784537436111372, 5.15952107035195224876390695669, 6.27535678684710522357357024314, 6.89481145635884210142070041791, 7.57732331310658119779307421080, 8.258048378412766658462105824666, 9.110000778761161927338246029646

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