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Properties

Label 2-945-5.4-c1-0-16
Degree $2$
Conductor $945$
Sign $-0.978 - 0.204i$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
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  + 2.18i·2-s − 2.79·4-s + (2.18 + 0.456i)5-s + i·7-s − 1.73i·8-s + (−0.999 + 4.79i)10-s + 1.73·11-s + 4.79i·13-s − 2.18·14-s − 1.79·16-s − 5.65i·17-s + 6.79·19-s + (−6.10 − 1.27i)20-s + 3.79i·22-s + 4.83i·23-s + ⋯
L(s)  = 1  + 1.54i·2-s − 1.39·4-s + (0.978 + 0.204i)5-s + 0.377i·7-s − 0.612i·8-s + (−0.316 + 1.51i)10-s + 0.522·11-s + 1.32i·13-s − 0.585·14-s − 0.447·16-s − 1.37i·17-s + 1.55·19-s + (−1.36 − 0.285i)20-s + 0.808i·22-s + 1.00i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.978 - 0.204i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.978 - 0.204i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.183952 + 1.78173i\)
\(L(\frac12)\) \(\approx\) \(0.183952 + 1.78173i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.18 - 0.456i)T \)
7 \( 1 - iT \)
good2 \( 1 - 2.18iT - 2T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 - 4.79iT - 13T^{2} \)
17 \( 1 + 5.65iT - 17T^{2} \)
19 \( 1 - 6.79T + 19T^{2} \)
23 \( 1 - 4.83iT - 23T^{2} \)
29 \( 1 - 3.00T + 29T^{2} \)
31 \( 1 + 8.58T + 31T^{2} \)
37 \( 1 - 10.1iT - 37T^{2} \)
41 \( 1 + 9.11T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 1.73iT - 47T^{2} \)
53 \( 1 + 6.56iT - 53T^{2} \)
59 \( 1 + 7.74T + 59T^{2} \)
61 \( 1 - 4.79T + 61T^{2} \)
67 \( 1 + 1.37iT - 67T^{2} \)
71 \( 1 - 1.17T + 71T^{2} \)
73 \( 1 + 4.58iT - 73T^{2} \)
79 \( 1 + 3.37T + 79T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 - 8.66T + 89T^{2} \)
97 \( 1 - 1.37iT - 97T^{2} \)
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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

−9.946210416904947886659943418204, −9.285299980212244905657695405305, −8.888041750523860595397420983239, −7.60545138637303909155623167090, −6.93598913264344156759503374816, −6.32622768536883538669897581529, −5.34671325681116239254818031478, −4.84103183702620929939732749906, −3.24131273684592580492800073099, −1.74186733076079139804656394510, 0.902267585546710843733223853518, 1.90916921526486606349985009881, 3.06679691119750474647143765210, 3.90485548420766991901229049357, 5.10347683340262460186848219220, 5.96823054959308293016664039607, 7.12475426757440129105613010143, 8.355598914310530176143858037803, 9.181669731725352567909307609091, 9.928503009243101358688285162769

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