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Properties

Label 2-1368-19.7-c1-0-14
Degree $2$
Conductor $1368$
Sign $0.624 + 0.781i$
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  + (−1.87 − 3.25i)5-s + 4.75·7-s + 4.36·11-s + (−0.5 + 0.866i)13-s + (3.06 + 5.30i)17-s + (0.694 − 4.30i)19-s + (−1.87 + 3.25i)23-s + (−4.56 + 7.90i)25-s + (2.69 − 4.66i)29-s + 7.36·31-s + (−8.94 − 15.4i)35-s − 7.12·37-s + (−1.75 − 3.04i)41-s + (0.379 + 0.657i)43-s + (−3 + 5.19i)47-s + ⋯
L(s)  = 1  + (−0.840 − 1.45i)5-s + 1.79·7-s + 1.31·11-s + (−0.138 + 0.240i)13-s + (0.743 + 1.28i)17-s + (0.159 − 0.987i)19-s + (−0.391 + 0.678i)23-s + (−0.912 + 1.58i)25-s + (0.500 − 0.866i)29-s + 1.32·31-s + (−1.51 − 2.61i)35-s − 1.17·37-s + (−0.274 − 0.475i)41-s + (0.0578 + 0.100i)43-s + (−0.437 + 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.624 + 0.781i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ 0.624 + 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.945080170\)
\(L(\frac12)\) \(\approx\) \(1.945080170\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (-0.694 + 4.30i)T \)
good5 \( 1 + (1.87 + 3.25i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 4.75T + 7T^{2} \)
11 \( 1 - 4.36T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.06 - 5.30i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.87 - 3.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.69 + 4.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.36T + 31T^{2} \)
37 \( 1 + 7.12T + 37T^{2} \)
41 \( 1 + (1.75 + 3.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.379 - 0.657i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.57 + 4.45i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.63 - 8.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.25 + 7.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.684 - 1.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.82 + 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.25 + 3.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.07 + 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.90T + 83T^{2} \)
89 \( 1 + (-4.18 + 7.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.82 - 10.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.178724253221013078920270372312, −8.530259303339463906471112908466, −8.097257066277226682124131724611, −7.30479455571430664432478451628, −6.02985381817192567098938745587, −4.98355793472549819679874330248, −4.47288433553394648561383533638, −3.73283721723274827514032231770, −1.76666267177002707174778107922, −1.02771656043443994500905779604, 1.28898180626818330580976499530, 2.64228846187652217405153783509, 3.66318181835504861490790906913, 4.50124359781183489364474120124, 5.49369817148241165793102213369, 6.71694453175250487800194156024, 7.23538270313467435043580921827, 8.074954826826714272107238239848, 8.602999749841791683827203685452, 9.972160998576949139109787076720

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