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Properties

Label 2-945-63.59-c1-0-3
Degree $2$
Conductor $945$
Sign $0.519 - 0.854i$
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  + (−1.87 − 1.08i)2-s + (1.33 + 2.31i)4-s − 5-s + (−1.75 − 1.98i)7-s − 1.46i·8-s + (1.87 + 1.08i)10-s − 1.72i·11-s + (−2.89 − 1.66i)13-s + (1.13 + 5.60i)14-s + (1.09 − 1.89i)16-s + (−2.47 + 4.28i)17-s + (2.88 − 1.66i)19-s + (−1.33 − 2.31i)20-s + (−1.86 + 3.23i)22-s − 2.67i·23-s + ⋯
L(s)  = 1  + (−1.32 − 0.764i)2-s + (0.669 + 1.15i)4-s − 0.447·5-s + (−0.661 − 0.749i)7-s − 0.517i·8-s + (0.592 + 0.341i)10-s − 0.520i·11-s + (−0.801 − 0.462i)13-s + (0.303 + 1.49i)14-s + (0.273 − 0.474i)16-s + (−0.600 + 1.04i)17-s + (0.662 − 0.382i)19-s + (−0.299 − 0.518i)20-s + (−0.398 + 0.689i)22-s − 0.557i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.519 - 0.854i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.519 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171586 + 0.0964624i\)
\(L(\frac12)\) \(\approx\) \(0.171586 + 0.0964624i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (1.75 + 1.98i)T \)
good2 \( 1 + (1.87 + 1.08i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + 1.72iT - 11T^{2} \)
13 \( 1 + (2.89 + 1.66i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.47 - 4.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.88 + 1.66i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.67iT - 23T^{2} \)
29 \( 1 + (-4.72 + 2.72i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.73 - 3.89i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.70 + 2.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.394 + 0.683i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.82 - 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.76 - 9.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.69 + 3.28i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.45 - 9.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.58 + 3.22i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.06 - 1.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.34iT - 71T^{2} \)
73 \( 1 + (-2.68 - 1.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.80 - 11.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.95 + 8.58i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.36 - 9.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.88 + 1.66i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.17410415793941796394988769380, −9.466140757269125567147875888931, −8.661188924263563145834563824199, −7.85693421460736961636856395001, −7.18810745042952627381673677273, −6.11742652630146628437611823818, −4.70693497389857102231047674547, −3.49345825747183981503100554248, −2.60235828646451563396243794782, −1.05078544288814221708344175591, 0.17010337986815760555448678615, 2.02322614742403619757088479925, 3.41414335084108210501804427388, 4.84925960400105641262630306722, 5.86519791062513042446546540100, 7.00910558395050003313154654572, 7.24212456775186596551909966720, 8.310010246735511866816636986436, 9.147906632126614371595422238609, 9.567432594245536166735862167479

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