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Properties

Label 2-945-21.20-c1-0-19
Degree $2$
Conductor $945$
Sign $0.959 - 0.282i$
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  − 0.422i·2-s + 1.82·4-s − 5-s + (2.53 − 0.746i)7-s − 1.61i·8-s + 0.422i·10-s + 5.39i·11-s + 6.54i·13-s + (−0.315 − 1.07i)14-s + 2.95·16-s − 0.230·17-s + 1.27i·19-s − 1.82·20-s + 2.28·22-s + 5.57i·23-s + ⋯
L(s)  = 1  − 0.298i·2-s + 0.910·4-s − 0.447·5-s + (0.959 − 0.282i)7-s − 0.571i·8-s + 0.133i·10-s + 1.62i·11-s + 1.81i·13-s + (−0.0843 − 0.286i)14-s + 0.739·16-s − 0.0560·17-s + 0.293i·19-s − 0.407·20-s + 0.486·22-s + 1.16i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.959 - 0.282i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.959 - 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99968 + 0.287819i\)
\(L(\frac12)\) \(\approx\) \(1.99968 + 0.287819i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.53 + 0.746i)T \)
good2 \( 1 + 0.422iT - 2T^{2} \)
11 \( 1 - 5.39iT - 11T^{2} \)
13 \( 1 - 6.54iT - 13T^{2} \)
17 \( 1 + 0.230T + 17T^{2} \)
19 \( 1 - 1.27iT - 19T^{2} \)
23 \( 1 - 5.57iT - 23T^{2} \)
29 \( 1 + 5.97iT - 29T^{2} \)
31 \( 1 + 8.57iT - 31T^{2} \)
37 \( 1 + 4.06T + 37T^{2} \)
41 \( 1 - 8.85T + 41T^{2} \)
43 \( 1 + 0.350T + 43T^{2} \)
47 \( 1 + 5.42T + 47T^{2} \)
53 \( 1 + 3.92iT - 53T^{2} \)
59 \( 1 - 9.14T + 59T^{2} \)
61 \( 1 - 8.57iT - 61T^{2} \)
67 \( 1 - 6.96T + 67T^{2} \)
71 \( 1 + 0.646iT - 71T^{2} \)
73 \( 1 - 1.00iT - 73T^{2} \)
79 \( 1 + 6.91T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + 6.54T + 89T^{2} \)
97 \( 1 + 9.40iT - 97T^{2} \)
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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

−10.01513165701960731958920116276, −9.539127999434622289962838779664, −8.257700541527646228636446175129, −7.31689112046845163944492398170, −7.04767746020995850590806131208, −5.81272733561717941895521596267, −4.47352789758418614980779775235, −3.97151496869980229352307582474, −2.27053776804995678884305850038, −1.61037698457360669398522777209, 1.02223318626394998372268589904, 2.66789716732488125083527169558, 3.42175464457918597966642885138, 5.04084553348582697091793532665, 5.64301238872531103976796876477, 6.60459095738907739275497124223, 7.59534719333372896458245432668, 8.339680333702331747916002633237, 8.686552718900212076679966140376, 10.43615468448776063969320875745

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