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Properties

Label 2-3120-3120.1949-c0-0-7
Degree $2$
Conductor $3120$
Sign $-1$
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  + (0.382 − 0.923i)2-s + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)4-s + (0.382 − 0.923i)5-s + (−0.382 − 0.923i)6-s + (−0.923 + 0.382i)8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s + (−0.541 + 0.541i)11-s − 12-s + (0.707 − 0.707i)13-s + (−0.382 − 0.923i)15-s + i·16-s + (−0.923 − 0.382i)18-s + (−0.923 + 0.382i)20-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)4-s + (0.382 − 0.923i)5-s + (−0.382 − 0.923i)6-s + (−0.923 + 0.382i)8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s + (−0.541 + 0.541i)11-s − 12-s + (0.707 − 0.707i)13-s + (−0.382 − 0.923i)15-s + i·16-s + (−0.923 − 0.382i)18-s + (−0.923 + 0.382i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.769073665\)
\(L(\frac12)\) \(\approx\) \(1.769073665\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 + 0.923i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 0.765iT - T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + 0.765iT - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
61 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.84iT - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
89 \( 1 + 1.84iT - T^{2} \)
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

−8.616842813441395918921641173796, −8.043958633941458926784267303942, −7.03916832657744474609294925623, −5.97859820766816457473255223358, −5.40810073531281050752182148829, −4.45516038274599375007924922225, −3.61068569309477342400799265340, −2.65751067717088837546318470689, −1.83861929227473687959632855904, −0.894529019351686759719643734414, 2.14067373314461147337936467743, 3.18721067119952693870857081655, 3.67585952640545433457550320202, 4.66540275532470518127773531113, 5.46573218253460230872548727168, 6.27601947364244501611388792491, 6.91466508760672621159598225268, 7.88494962168194797769393423642, 8.280452811331260498399245383085, 9.290276236726941505740789111889

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