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Properties

Label 2-945-35.13-c1-0-22
Degree $2$
Conductor $945$
Sign $-0.787 - 0.615i$
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  + (−0.834 + 0.834i)2-s + 0.606i·4-s + (1.65 + 1.50i)5-s + (−0.411 + 2.61i)7-s + (−2.17 − 2.17i)8-s + (−2.63 + 0.121i)10-s + 5.36·11-s + (0.211 − 0.211i)13-s + (−1.83 − 2.52i)14-s + 2.41·16-s + (4.64 + 4.64i)17-s − 4.27·19-s + (−0.914 + 1.00i)20-s + (−4.48 + 4.48i)22-s + (4.44 + 4.44i)23-s + ⋯
L(s)  = 1  + (−0.590 + 0.590i)2-s + 0.303i·4-s + (0.738 + 0.673i)5-s + (−0.155 + 0.987i)7-s + (−0.769 − 0.769i)8-s + (−0.833 + 0.0384i)10-s + 1.61·11-s + (0.0585 − 0.0585i)13-s + (−0.491 − 0.674i)14-s + 0.604·16-s + (1.12 + 1.12i)17-s − 0.979·19-s + (−0.204 + 0.224i)20-s + (−0.955 + 0.955i)22-s + (0.926 + 0.926i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.787 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.787 - 0.615i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.787 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.432730 + 1.25608i\)
\(L(\frac12)\) \(\approx\) \(0.432730 + 1.25608i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 - 1.50i)T \)
7 \( 1 + (0.411 - 2.61i)T \)
good2 \( 1 + (0.834 - 0.834i)T - 2iT^{2} \)
11 \( 1 - 5.36T + 11T^{2} \)
13 \( 1 + (-0.211 + 0.211i)T - 13iT^{2} \)
17 \( 1 + (-4.64 - 4.64i)T + 17iT^{2} \)
19 \( 1 + 4.27T + 19T^{2} \)
23 \( 1 + (-4.44 - 4.44i)T + 23iT^{2} \)
29 \( 1 + 4.03iT - 29T^{2} \)
31 \( 1 + 6.55iT - 31T^{2} \)
37 \( 1 + (4.85 - 4.85i)T - 37iT^{2} \)
41 \( 1 + 1.91iT - 41T^{2} \)
43 \( 1 + (-5.01 - 5.01i)T + 43iT^{2} \)
47 \( 1 + (7.26 + 7.26i)T + 47iT^{2} \)
53 \( 1 + (1.40 + 1.40i)T + 53iT^{2} \)
59 \( 1 + 1.34T + 59T^{2} \)
61 \( 1 + 1.03iT - 61T^{2} \)
67 \( 1 + (-7.23 + 7.23i)T - 67iT^{2} \)
71 \( 1 + 2.92T + 71T^{2} \)
73 \( 1 + (9.72 - 9.72i)T - 73iT^{2} \)
79 \( 1 - 2.78iT - 79T^{2} \)
83 \( 1 + (-4.47 + 4.47i)T - 83iT^{2} \)
89 \( 1 + 2.53T + 89T^{2} \)
97 \( 1 + (13.1 + 13.1i)T + 97iT^{2} \)
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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

−9.960553390003151879009517321760, −9.461227201137264676505543276933, −8.727428539163205948947183052569, −7.962753579267436350866588340510, −6.84674905986555398137261658486, −6.28926471746051923303786493565, −5.62878440044215939393791584945, −3.92668813756550447382160309009, −3.06171514000768378825736786448, −1.68761107840207123076484263790, 0.838770740978870378276051698976, 1.59555826859196667830254598823, 3.07127256169582428605414429204, 4.38132393670325707499561448770, 5.29991578286401185157510358716, 6.37330151597514030085406066487, 7.05717558693192449692428394363, 8.476833776330125951492904339376, 9.123698513084846887597013196849, 9.658610108029795167064934643749

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