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Properties

Label 2-2120-2120.317-c0-0-0
Degree $2$
Conductor $2120$
Sign $0.973 + 0.229i$
Analytic cond. $1.05801$
Root an. cond. $1.02859$
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  + (0.707 + 0.707i)2-s + (−1.41 + 1.41i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s − 2.00·6-s + (−1 − i)7-s + (−0.707 + 0.707i)8-s − 3.00i·9-s + 1.00·10-s + (−1.41 − 1.41i)12-s − 1.41i·14-s + 2.00i·15-s − 1.00·16-s + (−1 − i)17-s + (2.12 − 2.12i)18-s − 1.41i·19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−1.41 + 1.41i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s − 2.00·6-s + (−1 − i)7-s + (−0.707 + 0.707i)8-s − 3.00i·9-s + 1.00·10-s + (−1.41 − 1.41i)12-s − 1.41i·14-s + 2.00i·15-s − 1.00·16-s + (−1 − i)17-s + (2.12 − 2.12i)18-s − 1.41i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2120\)    =    \(2^{3} \cdot 5 \cdot 53\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(1.05801\)
Root analytic conductor: \(1.02859\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2120} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2120,\ (\ :0),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6310379458\)
\(L(\frac12)\) \(\approx\) \(0.6310379458\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
7 \( 1 + (1 + i)T + iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−9.290312192840133844527898497701, −8.849830878676702770400700631184, −7.21109097804907032615542183665, −6.54429670636421316104179274616, −6.08521568343809063152214536181, −5.00322436537271352260514757553, −4.75312624334806365267076648480, −3.93438864959508059739449142373, −2.97408879401533061341151497796, −0.40124911765435560377601349650, 1.62850064485113365967521318389, 2.20759381300570356902980962090, 3.24050446981285120949565517736, 4.72344808314074669415318623964, 5.70608221520693408555721347467, 6.20568999335461215660493781035, 6.36255438184508091917994091035, 7.35929341212984009466342250237, 8.529679084551758797084930412608, 9.689167431289596481125318229390

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