Properties

Label 2-1792-28.27-c1-0-33
Degree $2$
Conductor $1792$
Sign $-i$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.97·3-s + 4.29i·5-s + 2.64i·7-s + 5.82·9-s − 0.737i·13-s + 12.7i·15-s + 5.43·19-s + 7.86i·21-s − 7.48i·23-s − 13.4·25-s + 8.40·27-s − 11.3·35-s − 2.19i·39-s + 25.0i·45-s − 7.00·49-s + ⋯
L(s)  = 1  + 1.71·3-s + 1.92i·5-s + 0.999i·7-s + 1.94·9-s − 0.204i·13-s + 3.29i·15-s + 1.24·19-s + 1.71i·21-s − 1.56i·23-s − 2.69·25-s + 1.61·27-s − 1.92·35-s − 0.350i·39-s + 3.73i·45-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.263596351\)
\(L(\frac12)\) \(\approx\) \(3.263596351\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - 2.64iT \)
good3 \( 1 - 2.97T + 3T^{2} \)
5 \( 1 - 4.29iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 0.737iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 5.43T + 19T^{2} \)
23 \( 1 + 7.48iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 6.45T + 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 15.8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 5.29iT - 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.568342859241745646133180322949, −8.559531836401813859964668325020, −8.006934958170694978279504645006, −7.17270402553896955660716086765, −6.60107771026634160166014110154, −5.55199644202201308678503050667, −4.11720908732470432025275307726, −3.13378362240551053303312175677, −2.77878772948080805674055621091, −2.00501283909746227249308836956, 1.04628088297666194502980030677, 1.79697438066615160203114329376, 3.27192836174525397274491024192, 3.99112206991323480395948132801, 4.72163125184598627652589259433, 5.63651310704775191864978457804, 7.19319153841218235610179471367, 7.72122650694593350952267488283, 8.331113352560482452035066856865, 9.117608680321717648873554275933

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