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Properties

Label 2-1368-152.75-c1-0-28
Degree $2$
Conductor $1368$
Sign $0.999 + 0.0136i$
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  + (−1.33 − 0.452i)2-s + (1.59 + 1.21i)4-s − 0.393i·5-s + 1.35i·7-s + (−1.58 − 2.34i)8-s + (−0.177 + 0.526i)10-s − 1.21·11-s − 3.30·13-s + (0.612 − 1.81i)14-s + (1.05 + 3.85i)16-s + 0.00555·17-s + (2.48 − 3.58i)19-s + (0.476 − 0.625i)20-s + (1.62 + 0.550i)22-s + 0.677i·23-s + ⋯
L(s)  = 1  + (−0.947 − 0.320i)2-s + (0.795 + 0.606i)4-s − 0.175i·5-s + 0.511i·7-s + (−0.559 − 0.829i)8-s + (−0.0562 + 0.166i)10-s − 0.366·11-s − 0.917·13-s + (0.163 − 0.484i)14-s + (0.264 + 0.964i)16-s + 0.00134·17-s + (0.570 − 0.821i)19-s + (0.106 − 0.139i)20-s + (0.347 + 0.117i)22-s + 0.141i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.999 + 0.0136i$
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: \((2,\ 1368,\ (\ :1/2),\ 0.999 + 0.0136i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9837586150\)
\(L(\frac12)\) \(\approx\) \(0.9837586150\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.452i)T \)
3 \( 1 \)
19 \( 1 + (-2.48 + 3.58i)T \)
good5 \( 1 + 0.393iT - 5T^{2} \)
7 \( 1 - 1.35iT - 7T^{2} \)
11 \( 1 + 1.21T + 11T^{2} \)
13 \( 1 + 3.30T + 13T^{2} \)
17 \( 1 - 0.00555T + 17T^{2} \)
23 \( 1 - 0.677iT - 23T^{2} \)
29 \( 1 - 5.04T + 29T^{2} \)
31 \( 1 - 4.84T + 31T^{2} \)
37 \( 1 + 4.23T + 37T^{2} \)
41 \( 1 + 1.96iT - 41T^{2} \)
43 \( 1 - 4.59T + 43T^{2} \)
47 \( 1 - 10.9iT - 47T^{2} \)
53 \( 1 - 4.67T + 53T^{2} \)
59 \( 1 - 9.88iT - 59T^{2} \)
61 \( 1 + 4.99iT - 61T^{2} \)
67 \( 1 - 2.46iT - 67T^{2} \)
71 \( 1 + 0.424T + 71T^{2} \)
73 \( 1 + 1.23T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 7.42T + 83T^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 - 2.05iT - 97T^{2} \)
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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

−9.462252160005973923529515576456, −8.931060329171686925582169468908, −8.099684789170785099635624995636, −7.31540695644164246273401287935, −6.56886363814387077510386898732, −5.45029741234128941695647994304, −4.48689508664498858010492870766, −3.02612385025551004448804663410, −2.38510443324440443438186837372, −0.884521177993313181158894401448, 0.76962340510761580174360283427, 2.19378962763661416665214127522, 3.24677630108693396701204700428, 4.69666134707311922359159184010, 5.55970711694384130798155373453, 6.61728535317722897423555455493, 7.22709710933124895611956720688, 7.983964586012489658015247734644, 8.710105558129999863933285810555, 9.683302014274831183610661513608

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