Properties

Label 2-945-315.139-c0-0-1
Degree $2$
Conductor $945$
Sign $0.642 + 0.766i$
Analytic cond. $0.471616$
Root an. cond. $0.686743$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 17-s + (0.499 + 0.866i)20-s + (−0.499 − 0.866i)25-s + 0.999·28-s + (1 + 1.73i)29-s − 0.999·35-s + 0.999·44-s + (−0.5 − 0.866i)47-s + (−0.499 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 17-s + (0.499 + 0.866i)20-s + (−0.499 − 0.866i)25-s + 0.999·28-s + (1 + 1.73i)29-s − 0.999·35-s + 0.999·44-s + (−0.5 − 0.866i)47-s + (−0.499 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(0.471616\)
Root analytic conductor: \(0.686743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9000004152\)
\(L(\frac12)\) \(\approx\) \(0.9000004152\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.11901048241373687820457123035, −9.205925282059142179978326129197, −8.384684693344255517241302543645, −7.892317866102501250990284685082, −6.79834002760177459808336535315, −5.64552044558599080666745094203, −4.88927751558656183274434647864, −3.70209033157564122148272862830, −3.00602094776612171978053568942, −0.952758889404859546188942638999, 1.80642000281342737241429253877, 2.83762131276282827919223607362, 4.21255227783365436383376621636, 5.31468164366994026288209638955, 6.09350537486027703388368163867, 6.68105614222819842150755975577, 7.87142172923039882158240569249, 8.949099719176624520828267861193, 9.835433203939503953090709192534, 9.984144676664577140897230217441

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