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Properties

Label 2-3120-3120.1907-c0-0-0
Degree $2$
Conductor $3120$
Sign $-0.994 - 0.107i$
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 + (0.866 + 0.5i)3-s + (0.866 − 0.499i)4-s + (−0.707 + 0.707i)5-s + (−0.965 − 0.258i)6-s + (−0.366 + 1.36i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.500 − 0.866i)10-s + (0.258 + 0.965i)11-s + 12-s − 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 + (0.866 + 0.5i)3-s + (0.866 − 0.499i)4-s + (−0.707 + 0.707i)5-s + (−0.965 − 0.258i)6-s + (−0.366 + 1.36i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.500 − 0.866i)10-s + (0.258 + 0.965i)11-s + 12-s − 13-s − 1.41i·14-s + (−0.965 + 0.258i)15-s + (0.500 − 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.994 - 0.107i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1907, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ -0.994 - 0.107i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7457056772\)
\(L(\frac12)\) \(\approx\) \(0.7457056772\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + T \)
good7 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
53 \( 1 - 1.41T + T^{2} \)
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

−9.319789537148675581436152661615, −8.525689847542545208753221945840, −7.84778990824172288610170189389, −7.18448748158621747658778584765, −6.59158090242961266080900868888, −5.46095589372844508041424149633, −4.63824744366246467160282491723, −3.40698037087423882362422099014, −2.59874670046216934930339789848, −2.03555003694657285273094622999, 0.56120001942378627365075858366, 1.44812637062162576588901128241, 2.79762659237484897332757441839, 3.61038368547975263817004002853, 4.16861934120580355255367495879, 5.57249363529372634371571337389, 6.92091167244687575262621712011, 7.17036860069806380545577678604, 7.79778968903468490462808758725, 8.598108421612077897171727644080

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