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Properties

Label 2-945-5.4-c1-0-17
Degree $2$
Conductor $945$
Sign $0.204 - 0.978i$
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  + 0.456i·2-s + 1.79·4-s + (−0.456 + 2.18i)5-s − i·7-s + 1.73i·8-s + (−0.999 − 0.208i)10-s + 1.73·11-s − 0.208i·13-s + 0.456·14-s + 2.79·16-s + 3.00i·17-s + 2.20·19-s + (−0.818 + 3.92i)20-s + 0.791i·22-s + 3.10i·23-s + ⋯
L(s)  = 1  + 0.323i·2-s + 0.895·4-s + (−0.204 + 0.978i)5-s − 0.377i·7-s + 0.612i·8-s + (−0.316 − 0.0660i)10-s + 0.522·11-s − 0.0578i·13-s + 0.122·14-s + 0.697·16-s + 0.729i·17-s + 0.506·19-s + (−0.182 + 0.876i)20-s + 0.168i·22-s + 0.646i·23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.204 - 0.978i$
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: \((2,\ 945,\ (\ :1/2),\ 0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51296 + 1.22978i\)
\(L(\frac12)\) \(\approx\) \(1.51296 + 1.22978i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.456 - 2.18i)T \)
7 \( 1 + iT \)
good2 \( 1 - 0.456iT - 2T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 + 0.208iT - 13T^{2} \)
17 \( 1 - 3.00iT - 17T^{2} \)
19 \( 1 - 2.20T + 19T^{2} \)
23 \( 1 - 3.10iT - 23T^{2} \)
29 \( 1 - 5.65T + 29T^{2} \)
31 \( 1 - 0.582T + 31T^{2} \)
37 \( 1 - 8.16iT - 37T^{2} \)
41 \( 1 + 6.47T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + 1.73iT - 47T^{2} \)
53 \( 1 + 1.37iT - 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 - 0.208T + 61T^{2} \)
67 \( 1 + 12.3iT - 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + 4.58iT - 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 6.83iT - 83T^{2} \)
89 \( 1 - 8.66T + 89T^{2} \)
97 \( 1 - 12.3iT - 97T^{2} \)
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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

−10.40162314953979446567299828660, −9.549985628350547694756902437871, −8.200418296027253919520448531259, −7.66458665419436299776311687278, −6.61432591088390714683203355920, −6.39610212058854952942345196465, −5.11605271144062393343931281547, −3.72503868177621910729527527071, −2.92243670996024171884095627553, −1.60883210237054922634982649309, 0.972420488616498831440526230080, 2.21497719604253992350322779417, 3.37335262069966511475326813110, 4.51542583911330078155634383489, 5.50199672602639983408678496720, 6.45559798167401767684262435593, 7.33510543513732264243742411312, 8.246299532839084818803423509276, 9.107425287089104097953038204547, 9.823451266697743417535511845139

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