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Properties

Label 2-945-945.209-c1-0-15
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

\(L(1)\) \(\approx\) \(0.105057 - 0.691578i\)
\(L(\frac12)\) \(\approx\) \(0.105057 - 0.691578i\)
\(L(\frac{3}{2})\) 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

−10.34353952420698816522356032914, −9.689503930063603770877915419526, −8.776607337655202679640347395500, −8.317818654873006277353881028411, −7.44220580321086673782113063297, −5.71587040839371798639772592620, −4.99560098346524518695616473014, −4.76032769863890940747118066255, −3.01486409200343825695511826771, −2.43622834107540227937082245574, 0.29553980094554819147203762741, 1.92371362914940133924075107597, 2.58334698933582002803336875163, 4.63704525301260195261834279969, 5.12573142563484701606975690047, 6.11132241424702674069425961036, 7.15901918909341485582233730483, 7.81167750228170322475968296577, 8.712922710271050856333102320006, 9.764124045311164428685489520375

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