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Properties

Label 2-912-228.203-c0-0-0
Degree $2$
Conductor $912$
Sign $0.939 - 0.341i$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
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  + (0.939 + 0.342i)3-s + (0.592 + 0.342i)7-s + (0.766 + 0.642i)9-s + (−0.673 − 1.85i)13-s + (−0.5 + 0.866i)19-s + (0.439 + 0.524i)21-s + (−0.939 + 0.342i)25-s + (0.500 + 0.866i)27-s + (−0.766 + 1.32i)31-s − 1.28i·37-s − 1.96i·39-s + (1.26 − 0.223i)43-s + (−0.266 − 0.460i)49-s + (−0.766 + 0.642i)57-s + (−0.0603 + 0.342i)61-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)3-s + (0.592 + 0.342i)7-s + (0.766 + 0.642i)9-s + (−0.673 − 1.85i)13-s + (−0.5 + 0.866i)19-s + (0.439 + 0.524i)21-s + (−0.939 + 0.342i)25-s + (0.500 + 0.866i)27-s + (−0.766 + 1.32i)31-s − 1.28i·37-s − 1.96i·39-s + (1.26 − 0.223i)43-s + (−0.266 − 0.460i)49-s + (−0.766 + 0.642i)57-s + (−0.0603 + 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.939 - 0.341i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :0),\ 0.939 - 0.341i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.384070644\)
\(L(\frac12)\) \(\approx\) \(1.384070644\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.592 - 0.342i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.673 + 1.85i)T + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.28iT - T^{2} \)
41 \( 1 + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (-1.26 + 0.223i)T + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (-1.11 - 1.32i)T + (-0.173 + 0.984i)T^{2} \)
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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

−10.37820068551894581864953608877, −9.450491808407271017378097342459, −8.623440884803313448409840325009, −7.86895553616979151193597810721, −7.35493272585046003837655889884, −5.79339885128572297127102923515, −5.07468819674546473997367138274, −3.91738372474238203419734475205, −2.95345671798628671305370679399, −1.85635878418563665278684712613, 1.69171350018321506822653094095, 2.59336203316051395121421551433, 4.08877437787564994918263989329, 4.57503025682112338142796732617, 6.15230474805707003614322782288, 7.07911050108522007826395800063, 7.65763222657011993622995948313, 8.628774856405525025769357910826, 9.324764582698478438317873409739, 9.993323383405854665179381323049

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