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Properties

Label 2-2856-2856.2603-c0-0-3
Degree $2$
Conductor $2856$
Sign $0.940 - 0.339i$
Analytic cond. $1.42532$
Root an. cond. $1.19387$
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 + (0.923 − 0.382i)3-s + 1.00i·4-s + (0.923 + 0.382i)6-s + (0.382 − 0.923i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.382 + 0.923i)12-s − 1.84i·13-s + (0.923 − 0.382i)14-s − 1.00·16-s + (0.382 + 0.923i)17-s + 18-s − i·21-s + (0.707 + 0.292i)23-s + (−0.382 + 0.923i)24-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (0.923 − 0.382i)3-s + 1.00i·4-s + (0.923 + 0.382i)6-s + (0.382 − 0.923i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (0.382 + 0.923i)12-s − 1.84i·13-s + (0.923 − 0.382i)14-s − 1.00·16-s + (0.382 + 0.923i)17-s + 18-s − i·21-s + (0.707 + 0.292i)23-s + (−0.382 + 0.923i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2856\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.940 - 0.339i$
Analytic conductor: \(1.42532\)
Root analytic conductor: \(1.19387\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2856} (2603, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2856,\ (\ :0),\ 0.940 - 0.339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.518562226\)
\(L(\frac12)\) \(\approx\) \(2.518562226\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (-0.382 - 0.923i)T \)
good5 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + 1.84iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
61 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
89 \( 1 - 0.765iT - T^{2} \)
97 \( 1 + (0.707 - 0.707i)T^{2} \)
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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

−8.568912314436586350216567675768, −7.984194274985523974717765195323, −7.64144168728619246931766900673, −6.87395398881692480346491855863, −5.97063918294314929022643838569, −5.18313147984113978309458477265, −4.15608607626479624080192367864, −3.48283298388376082723542575855, −2.77055592279304228955459266956, −1.34559612303449874539630150290, 1.74379294733722196958529002577, 2.30362707657068511302976639789, 3.26181070754884745054424773844, 4.11907367054592734200560222679, 4.85747029295889143906820027851, 5.50181937440685002746432409802, 6.66606494679026542717037956938, 7.29641918838897757229291172792, 8.611049397165197008427672871928, 8.952062371346040991220260612908

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