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Properties

Label 2-1368-152.75-c1-0-21
Degree $2$
Conductor $1368$
Sign $-0.900 - 0.434i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
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  + (1.26 + 0.639i)2-s + (1.18 + 1.61i)4-s + 1.96i·5-s + 1.25i·7-s + (0.458 + 2.79i)8-s + (−1.25 + 2.47i)10-s − 3.11·11-s − 0.401·13-s + (−0.803 + 1.58i)14-s + (−1.20 + 3.81i)16-s − 4.88·17-s + (−2.50 + 3.56i)19-s + (−3.16 + 2.32i)20-s + (−3.92 − 1.99i)22-s − 3.34i·23-s + ⋯
L(s)  = 1  + (0.891 + 0.452i)2-s + (0.590 + 0.806i)4-s + 0.878i·5-s + 0.474i·7-s + (0.162 + 0.986i)8-s + (−0.397 + 0.783i)10-s − 0.938·11-s − 0.111·13-s + (−0.214 + 0.423i)14-s + (−0.301 + 0.953i)16-s − 1.18·17-s + (−0.574 + 0.818i)19-s + (−0.708 + 0.519i)20-s + (−0.836 − 0.424i)22-s − 0.696i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.900 - 0.434i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ -0.900 - 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.204052769\)
\(L(\frac12)\) \(\approx\) \(2.204052769\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 0.639i)T \)
3 \( 1 \)
19 \( 1 + (2.50 - 3.56i)T \)
good5 \( 1 - 1.96iT - 5T^{2} \)
7 \( 1 - 1.25iT - 7T^{2} \)
11 \( 1 + 3.11T + 11T^{2} \)
13 \( 1 + 0.401T + 13T^{2} \)
17 \( 1 + 4.88T + 17T^{2} \)
23 \( 1 + 3.34iT - 23T^{2} \)
29 \( 1 - 3.79T + 29T^{2} \)
31 \( 1 - 2.06T + 31T^{2} \)
37 \( 1 - 1.55T + 37T^{2} \)
41 \( 1 - 2.92iT - 41T^{2} \)
43 \( 1 - 2.08T + 43T^{2} \)
47 \( 1 + 4.59iT - 47T^{2} \)
53 \( 1 - 1.83T + 53T^{2} \)
59 \( 1 - 7.40iT - 59T^{2} \)
61 \( 1 + 1.71iT - 61T^{2} \)
67 \( 1 - 10.2iT - 67T^{2} \)
71 \( 1 + 8.84T + 71T^{2} \)
73 \( 1 - 6.14T + 73T^{2} \)
79 \( 1 + 1.51T + 79T^{2} \)
83 \( 1 + 3.55T + 83T^{2} \)
89 \( 1 - 5.11iT - 89T^{2} \)
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

−10.25519992766071041986298796002, −8.832123461855380547680400785065, −8.204950410333535346995485455694, −7.28134691059168415251372544922, −6.55441242916819382116356158340, −5.89117937716187745069718587335, −4.90737792258560593946018711959, −4.04880312635088191196547911001, −2.83102240057284908507995194991, −2.30206307833544976257519175302, 0.62233755297059971028087949431, 2.03601768764633898063041045641, 3.05456788476918512173312242044, 4.36690288753988686789545226991, 4.75022395086410557972574151695, 5.68515855958770909063122150026, 6.65951001627312237470712950469, 7.47010368229001802105807756816, 8.545828488376590577206772561276, 9.320786670157055686945687924318

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