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Properties

Label 2-945-5.4-c1-0-5
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

\(L(1)\) \(\approx\) \(0.791823 - 0.0817508i\)
\(L(\frac12)\) \(\approx\) \(0.791823 - 0.0817508i\)
\(L(\frac{3}{2})\) 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

−10.20207816919851488727015013429, −9.309696547101802290896178234951, −8.673571352977644611896314935332, −7.71164555794783657843590205023, −6.68415756675247261719579996782, −5.30090904618887456190227142742, −4.26669587787356413789982273579, −3.61145116036443944254329650659, −2.50707273831073543360774532949, −1.30559380936349052883061588368, 0.41028041964600514500615788575, 2.94853352910896821568089382637, 3.98997775338137963929094197101, 5.30610879755805867387451918023, 5.54686870193906911163587397999, 7.11099508459148429100648709958, 7.46275728875334921659238389870, 7.894926219482424124809868473532, 9.004981576337246863076284936515, 9.781772414622468131269084579476

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