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Properties

Label 2-3120-3120.1043-c0-0-1
Degree $2$
Conductor $3120$
Sign $-0.778 - 0.627i$
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  + (−0.965 − 0.258i)2-s + (−0.5 − 0.866i)3-s + (0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (0.258 + 0.965i)6-s + (−0.366 − 1.36i)7-s + (−0.707 − 0.707i)8-s + (−0.499 + 0.866i)9-s + (−0.866 + 0.500i)10-s + (−0.965 − 0.258i)11-s − i·12-s − i·13-s + 1.41i·14-s + (−0.965 − 0.258i)15-s + (0.500 + 0.866i)16-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.5 − 0.866i)3-s + (0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (0.258 + 0.965i)6-s + (−0.366 − 1.36i)7-s + (−0.707 − 0.707i)8-s + (−0.499 + 0.866i)9-s + (−0.866 + 0.500i)10-s + (−0.965 − 0.258i)11-s − i·12-s − i·13-s + 1.41i·14-s + (−0.965 − 0.258i)15-s + (0.500 + 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.778 - 0.627i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ -0.778 - 0.627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4224253616\)
\(L(\frac12)\) \(\approx\) \(0.4224253616\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + iT \)
good7 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
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

−8.174332278476691813758049921923, −7.82363405040287164039747627991, −7.12119996957822516223429903735, −6.23426466023452328970251866172, −5.67704992427796142777817454456, −4.63479030314768775125890668045, −3.33349697394925499315382527545, −2.34953382139590935956635215608, −1.29752099452932883122082804271, −0.37470022237946577085710324937, 2.02094518189893321835550899041, 2.59385368366083507030341953558, 3.70134487541304804005621733216, 5.23300403087651287128768967900, 5.54181517284575613570493564243, 6.38020679974929825263718973600, 6.87744296137170326584886229911, 8.000986704678052552512506677815, 8.868820827975447006958200812021, 9.344241160518247309204511584872

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