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Properties

Label 2-1368-8.5-c1-0-71
Degree $2$
Conductor $1368$
Sign $0.666 + 0.745i$
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  + (1.37 − 0.340i)2-s + (1.76 − 0.935i)4-s − 2.13i·5-s + 3.29·7-s + (2.10 − 1.88i)8-s + (−0.727 − 2.93i)10-s + 3.71i·11-s + 2.32i·13-s + (4.52 − 1.12i)14-s + (2.25 − 3.30i)16-s + 6.48·17-s − i·19-s + (−1.99 − 3.77i)20-s + (1.26 + 5.09i)22-s − 7.32·23-s + ⋯
L(s)  = 1  + (0.970 − 0.240i)2-s + (0.883 − 0.467i)4-s − 0.954i·5-s + 1.24·7-s + (0.745 − 0.666i)8-s + (−0.229 − 0.926i)10-s + 1.11i·11-s + 0.645i·13-s + (1.20 − 0.299i)14-s + (0.562 − 0.826i)16-s + 1.57·17-s − 0.229i·19-s + (−0.446 − 0.843i)20-s + (0.269 + 1.08i)22-s − 1.52·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.666 + 0.745i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ 0.666 + 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.702569137\)
\(L(\frac12)\) \(\approx\) \(3.702569137\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.340i)T \)
3 \( 1 \)
19 \( 1 + iT \)
good5 \( 1 + 2.13iT - 5T^{2} \)
7 \( 1 - 3.29T + 7T^{2} \)
11 \( 1 - 3.71iT - 11T^{2} \)
13 \( 1 - 2.32iT - 13T^{2} \)
17 \( 1 - 6.48T + 17T^{2} \)
23 \( 1 + 7.32T + 23T^{2} \)
29 \( 1 - 2.59iT - 29T^{2} \)
31 \( 1 + 1.34T + 31T^{2} \)
37 \( 1 + 3.72iT - 37T^{2} \)
41 \( 1 + 6.52T + 41T^{2} \)
43 \( 1 + 1.97iT - 43T^{2} \)
47 \( 1 + 5.45T + 47T^{2} \)
53 \( 1 + 4.98iT - 53T^{2} \)
59 \( 1 + 9.67iT - 59T^{2} \)
61 \( 1 - 8.15iT - 61T^{2} \)
67 \( 1 + 0.524iT - 67T^{2} \)
71 \( 1 + 7.17T + 71T^{2} \)
73 \( 1 + 6.33T + 73T^{2} \)
79 \( 1 + 8.75T + 79T^{2} \)
83 \( 1 - 7.74iT - 83T^{2} \)
89 \( 1 - 1.04T + 89T^{2} \)
97 \( 1 + 0.117T + 97T^{2} \)
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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

−9.702179023835731670160635725445, −8.572317452950098777719331880627, −7.76449578074903069641431721942, −7.04596208391375788163038880289, −5.83985794398335681465738487873, −5.02399106403579920174578271495, −4.57629489712794185823285820108, −3.62231679783030384563979817639, −2.05594796207227322655878695765, −1.38028559332456452677021464260, 1.57837701464188591506430509037, 2.90375717751873832115547255548, 3.54813800620072866372619709622, 4.67356163505933126045379968868, 5.67337492751450464490035978283, 6.10182709690345185371704430710, 7.28490090775500938220713565411, 7.952905011908413659145281498906, 8.428028310959263587649279880242, 10.11300142384942882389213931221

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