Properties

Degree $2$
Conductor $1386$
Sign $0.839 + 0.544i$
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 + (−1.03 + 0.598i)5-s + (−2.52 − 0.782i)7-s + 0.999·8-s + (1.03 + 0.598i)10-s + (−0.396 + 3.29i)11-s − 3.26i·13-s + (0.586 + 2.57i)14-s + (−0.5 − 0.866i)16-s + (2.21 − 3.82i)17-s + (−6.73 + 3.88i)19-s − 1.19i·20-s + (3.04 − 1.30i)22-s + (5.94 − 3.43i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.463 + 0.267i)5-s + (−0.955 − 0.295i)7-s + 0.353·8-s + (0.327 + 0.189i)10-s + (−0.119 + 0.992i)11-s − 0.904i·13-s + (0.156 + 0.689i)14-s + (−0.125 − 0.216i)16-s + (0.536 − 0.928i)17-s + (−1.54 + 0.892i)19-s − 0.267i·20-s + (0.650 − 0.277i)22-s + (1.24 − 0.715i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.839 + 0.544i$
Motivic weight: \(1\)
Character: $\chi_{1386} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.839 + 0.544i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9513739201\)
\(L(\frac12)\) \(\approx\) \(0.9513739201\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.52 + 0.782i)T \)
11 \( 1 + (0.396 - 3.29i)T \)
good5 \( 1 + (1.03 - 0.598i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.26iT - 13T^{2} \)
17 \( 1 + (-2.21 + 3.82i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.73 - 3.88i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.94 + 3.43i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.85T + 29T^{2} \)
31 \( 1 + (0.103 - 0.179i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.83 - 6.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.15T + 41T^{2} \)
43 \( 1 + 8.00iT - 43T^{2} \)
47 \( 1 + (0.355 - 0.205i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.02 - 2.90i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.66 - 2.11i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.94 + 5.74i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.61 + 9.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.80iT - 71T^{2} \)
73 \( 1 + (0.195 + 0.113i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.00 - 1.15i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + (11.4 - 6.58i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−9.753071144215017269648697169465, −8.767139679086267296847309323516, −7.945272555651910192479761611250, −7.14792495079851633002769381015, −6.44283855602074307332872209313, −5.14883686355282530038169631308, −4.16453267922512272210951257013, −3.24477226205638835819232636577, −2.41734755221237295708476171906, −0.72908291865487526879093463648, 0.73314436195658965289784099080, 2.49416541741743349062406782881, 3.72055547916917017892066721672, 4.59042372515101493653296708100, 5.77842203911100924421781651126, 6.40724842008477697716809629343, 7.11255865806002764615882110328, 8.265843401685879375978821282939, 8.696060929045236048918519935654, 9.421520993588252130366387433498

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