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Properties

Label 2-1368-19.11-c1-0-15
Degree $2$
Conductor $1368$
Sign $0.910 + 0.412i$
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 − 1.73i)5-s + 3·7-s + 6·11-s + (0.5 + 0.866i)13-s + (1 − 1.73i)17-s + (−4 + 1.73i)19-s + (0.500 + 0.866i)25-s + (1 + 1.73i)29-s − 31-s + (3 − 5.19i)35-s − 7·37-s + (0.5 − 0.866i)43-s + 2·49-s + (2 + 3.46i)53-s + (6 − 10.3i)55-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + 1.13·7-s + 1.80·11-s + (0.138 + 0.240i)13-s + (0.242 − 0.420i)17-s + (−0.917 + 0.397i)19-s + (0.100 + 0.173i)25-s + (0.185 + 0.321i)29-s − 0.179·31-s + (0.507 − 0.878i)35-s − 1.15·37-s + (0.0762 − 0.132i)43-s + 0.285·49-s + (0.274 + 0.475i)53-s + (0.809 − 1.40i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.295364037\)
\(L(\frac12)\) \(\approx\) \(2.295364037\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.5 + 12.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.5 - 7.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.225625745886720171598510732026, −8.900400645572672298106023856480, −8.094166374400988116431568023200, −7.04571880996187750290626289361, −6.22687912254768452147324238119, −5.24568313561631342081293445516, −4.50668079186704118135919480250, −3.63118667314635298068719082385, −1.92137721713990289539439987616, −1.21663709903951145327394620484, 1.33633340468819628997221321433, 2.32184761195797885211662949271, 3.65555452104849576191233106448, 4.45821548773101626196950906517, 5.56703276868370906837850703674, 6.49966895804823778222046623604, 6.99302870625718669953158674130, 8.178746637089807812465327563453, 8.753297411822802100666184338691, 9.670489023489708317396972883091

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