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Properties

Label 2-945-945.209-c1-0-28
Degree $2$
Conductor $945$
Sign $0.954 - 0.296i$
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  + (−0.306 − 1.70i)3-s + (−1.53 + 1.28i)4-s + (−2.20 + 0.388i)5-s + (−2.02 − 1.70i)7-s + (−2.81 + 1.04i)9-s + (−6.48 − 1.14i)11-s + (2.66 + 2.21i)12-s + (5.77 + 2.10i)13-s + (1.33 + 3.63i)15-s + (0.694 − 3.93i)16-s + (2.93 − 1.69i)17-s + (2.87 − 3.42i)20-s + (−2.27 + 3.97i)21-s + (4.69 − 1.71i)25-s + (2.64 + 4.47i)27-s + 5.29·28-s + ⋯
L(s)  = 1  + (−0.177 − 0.984i)3-s + (−0.766 + 0.642i)4-s + (−0.984 + 0.173i)5-s + (−0.766 − 0.642i)7-s + (−0.937 + 0.348i)9-s + (−1.95 − 0.344i)11-s + (0.768 + 0.640i)12-s + (1.60 + 0.583i)13-s + (0.345 + 0.938i)15-s + (0.173 − 0.984i)16-s + (0.711 − 0.410i)17-s + (0.642 − 0.766i)20-s + (−0.496 + 0.867i)21-s + (0.939 − 0.342i)25-s + (0.509 + 0.860i)27-s + 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.954 - 0.296i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.954 - 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.569981 + 0.0865861i\)
\(L(\frac12)\) \(\approx\) \(0.569981 + 0.0865861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.306 + 1.70i)T \)
5 \( 1 + (2.20 - 0.388i)T \)
7 \( 1 + (2.02 + 1.70i)T \)
good2 \( 1 + (1.53 - 1.28i)T^{2} \)
11 \( 1 + (6.48 + 1.14i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (-5.77 - 2.10i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.93 + 1.69i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.67 - 10.1i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-3.77 + 4.50i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (14.5 - 8.38i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.258 + 0.447i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.82 + 1.75i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (1.01 + 2.79i)T + (-63.5 + 53.3i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.21 + 18.2i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.30294696756340910289438058352, −8.831673902476527471292882003974, −8.346272769082171010520422931540, −7.52924647745013382276554528144, −7.01366089167499881499787182947, −5.81978615132838462086073264734, −4.77808486820812826455475700471, −3.50168015832654263510797250683, −2.97574337900152040255476816870, −0.78320835233991045300103904336, 0.44443524795116619439854282962, 2.88400714008414450753869514743, 3.79449584061132744017422188461, 4.71471283838480814615003915615, 5.60647745267841998999487501006, 6.11838169210423577022738405875, 7.907596538558872437036278333191, 8.388909046837739762849520519110, 9.222552119326137962084671989171, 10.20274020478362922526435330718

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