Properties

Label 2-1944-1944.1771-c0-0-0
Degree $2$
Conductor $1944$
Sign $-0.280 - 0.959i$
Analytic cond. $0.970182$
Root an. cond. $0.984978$
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  + (0.249 − 0.968i)2-s + (−0.211 − 0.977i)3-s + (−0.875 − 0.483i)4-s + (−0.999 − 0.0387i)6-s + (−0.686 + 0.727i)8-s + (−0.910 + 0.413i)9-s + (−1.35 − 1.04i)11-s + (−0.286 + 0.957i)12-s + (0.533 + 0.845i)16-s + (−0.771 − 0.387i)17-s + (0.173 + 0.984i)18-s + (0.0157 − 0.270i)19-s + (−1.35 + 1.04i)22-s + (0.856 + 0.516i)24-s + (−0.565 + 0.824i)25-s + ⋯
L(s)  = 1  + (0.249 − 0.968i)2-s + (−0.211 − 0.977i)3-s + (−0.875 − 0.483i)4-s + (−0.999 − 0.0387i)6-s + (−0.686 + 0.727i)8-s + (−0.910 + 0.413i)9-s + (−1.35 − 1.04i)11-s + (−0.286 + 0.957i)12-s + (0.533 + 0.845i)16-s + (−0.771 − 0.387i)17-s + (0.173 + 0.984i)18-s + (0.0157 − 0.270i)19-s + (−1.35 + 1.04i)22-s + (0.856 + 0.516i)24-s + (−0.565 + 0.824i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.280 - 0.959i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (1771, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ -0.280 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4500008564\)
\(L(\frac12)\) \(\approx\) \(0.4500008564\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.249 + 0.968i)T \)
3 \( 1 + (0.211 + 0.977i)T \)
good5 \( 1 + (0.565 - 0.824i)T^{2} \)
7 \( 1 + (0.740 + 0.672i)T^{2} \)
11 \( 1 + (1.35 + 1.04i)T + (0.249 + 0.968i)T^{2} \)
13 \( 1 + (0.627 + 0.778i)T^{2} \)
17 \( 1 + (0.771 + 0.387i)T + (0.597 + 0.802i)T^{2} \)
19 \( 1 + (-0.0157 + 0.270i)T + (-0.993 - 0.116i)T^{2} \)
23 \( 1 + (0.211 + 0.977i)T^{2} \)
29 \( 1 + (0.981 - 0.192i)T^{2} \)
31 \( 1 + (0.790 + 0.612i)T^{2} \)
37 \( 1 + (0.286 + 0.957i)T^{2} \)
41 \( 1 + (1.05 - 1.03i)T + (0.0193 - 0.999i)T^{2} \)
43 \( 1 + (0.302 + 1.39i)T + (-0.910 + 0.413i)T^{2} \)
47 \( 1 + (0.790 - 0.612i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.144 + 1.05i)T + (-0.963 + 0.268i)T^{2} \)
61 \( 1 + (-0.533 + 0.845i)T^{2} \)
67 \( 1 + (0.0697 - 0.716i)T + (-0.981 - 0.192i)T^{2} \)
71 \( 1 + (0.835 + 0.549i)T^{2} \)
73 \( 1 + (-1.58 - 0.375i)T + (0.893 + 0.448i)T^{2} \)
79 \( 1 + (-0.856 + 0.516i)T^{2} \)
83 \( 1 + (1.41 + 1.39i)T + (0.0193 + 0.999i)T^{2} \)
89 \( 1 + (-0.0333 - 0.111i)T + (-0.835 + 0.549i)T^{2} \)
97 \( 1 + (-0.301 + 0.572i)T + (-0.565 - 0.824i)T^{2} \)
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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

−8.685707252229951991492235460842, −8.264876468248813765904916893895, −7.30602792829136690870335742270, −6.30502723618728198951246501463, −5.46052337072295782835451277231, −4.90791949276926320123266541412, −3.47684310788733830477252093445, −2.70605281676288582490735445865, −1.77267873582196003121664144183, −0.29214751300694434754645117651, 2.45530158632221433528707812238, 3.60641893670423531609274134836, 4.53541390780258783254652574261, 5.00245827245404963966193052454, 5.89229595854881131340856571480, 6.64169639578413922236356998439, 7.66511433705980260584645482862, 8.270512941131150808682649733669, 9.089182811806118216660166540523, 9.917404645211120498926845997629

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