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Properties

Label 2-1386-77.54-c1-0-30
Degree $2$
Conductor $1386$
Sign $0.477 - 0.878i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (2.53 + 1.46i)5-s + (2.00 + 1.72i)7-s + 0.999i·8-s + (1.46 + 2.53i)10-s + (0.608 − 3.26i)11-s + 1.09·13-s + (0.874 + 2.49i)14-s + (−0.5 + 0.866i)16-s + (2.00 + 3.47i)17-s + (3.32 − 5.76i)19-s + 2.92i·20-s + (2.15 − 2.51i)22-s + (0.874 − 1.51i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (1.13 + 0.653i)5-s + (0.758 + 0.652i)7-s + 0.353i·8-s + (0.461 + 0.800i)10-s + (0.183 − 0.983i)11-s + 0.302·13-s + (0.233 + 0.667i)14-s + (−0.125 + 0.216i)16-s + (0.485 + 0.841i)17-s + (0.763 − 1.32i)19-s + 0.653i·20-s + (0.459 − 0.537i)22-s + (0.182 − 0.315i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.289753323\)
\(L(\frac12)\) \(\approx\) \(3.289753323\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-2.00 - 1.72i)T \)
11 \( 1 + (-0.608 + 3.26i)T \)
good5 \( 1 + (-2.53 - 1.46i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.09T + 13T^{2} \)
17 \( 1 + (-2.00 - 3.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.32 + 5.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.874 + 1.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.40iT - 29T^{2} \)
31 \( 1 + (7.56 - 4.36i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.36 - 9.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.63T + 41T^{2} \)
43 \( 1 + 1.44iT - 43T^{2} \)
47 \( 1 + (1.67 + 0.969i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.32 + 5.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.85 + 3.95i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.37 + 2.38i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.70 + 2.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 + (-4.01 - 6.94i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.41 - 4.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.82T + 83T^{2} \)
89 \( 1 + (6.94 + 4.01i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−9.701516688980573783642334954100, −8.746596000246562413054695171633, −8.180738650123854401911030744800, −7.00066061202016091762291853466, −6.29779155260305307959637627819, −5.57552494913384614918734901718, −4.97182944531329414191323768887, −3.56213040270003539442802867754, −2.69795099220524541376389179656, −1.61787013567458554053888128440, 1.34124241641780068744101430433, 1.91132563113566361557950900990, 3.42378327455556665153675622376, 4.39874215891124749880746020585, 5.33900969010379995701369523397, 5.67878186092068394832787372215, 7.05314126309250411108199980126, 7.60745552032604752913759461064, 8.874121392502609775484625932304, 9.590133153843721770256950347688

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