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Properties

Label 2-1386-7.4-c1-0-29
Degree $2$
Conductor $1386$
Sign $-0.832 + 0.553i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
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  + (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

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06162074581\)
\(L(\frac12)\) \(\approx\) \(0.06162074581\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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