Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Sign $-0.832 - 0.553i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.60 + 2.77i)5-s + (1.60 − 2.10i)7-s − 0.999·8-s + (1.60 + 2.77i)10-s + (0.5 + 0.866i)11-s − 6.40·13-s + (−1.02 − 2.43i)14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.576 + 0.998i)19-s + 3.20·20-s + 0.999·22-s + (0.976 − 1.69i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.715 + 1.23i)5-s + (0.605 − 0.796i)7-s − 0.353·8-s + (0.506 + 0.876i)10-s + (0.150 + 0.261i)11-s − 1.77·13-s + (−0.273 − 0.652i)14-s + (−0.125 + 0.216i)16-s + (0.121 + 0.210i)17-s + (−0.132 + 0.229i)19-s + 0.715·20-s + 0.213·22-s + (0.203 − 0.352i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $-0.832 - 0.553i$
motivic weight  =  \(1\)
character  :  $\chi_{1386} (793, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1386,\ (\ :1/2),\ -0.832 - 0.553i)\)
\(L(1)\)  \(\approx\)  \(0.06162074581\)
\(L(\frac12)\)  \(\approx\)  \(0.06162074581\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-1.60 + 2.10i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (1.60 - 2.77i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 6.40T + 13T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.576 - 0.998i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.976 + 1.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.24T + 29T^{2} \)
31 \( 1 + (5.24 + 9.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.57 - 4.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.24T + 41T^{2} \)
43 \( 1 - 5.15T + 43T^{2} \)
47 \( 1 + (3.02 - 5.23i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.20 + 5.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.77 - 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.85 - 8.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.12 + 5.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.24T + 71T^{2} \)
73 \( 1 + (-1.04 - 1.81i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.80 + 4.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.55T + 83T^{2} \)
89 \( 1 + (9.20 - 15.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.49T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.11608346537372658011989714572, −9.417775114799458823778797691036, −8.065750949344605298243863250142, −7.36498044456750667675310403952, −6.89839465528464136846890152092, −5.61943375708091547241351174220, −4.55928918583613051558644840741, −3.88576389447668866589578455121, −2.90470292820128859807903846959, −1.88456246931673577459305186051, 0.02089584621481332332997949204, 1.85066185270786591950555710697, 3.27304800428326881138336107871, 4.43680300918022817724780497086, 5.13420629089399528464973569721, 5.53001806557814896893194490970, 7.00201561208835050330974655747, 7.60353177917116397505421678100, 8.450464519244387030409632480395, 8.990597061641364548778572954647

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