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Properties

Label 2-1386-33.8-c1-0-23
Degree $2$
Conductor $1386$
Sign $-0.748 + 0.662i$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (1.14 − 1.58i)5-s + (−0.951 − 0.309i)7-s + (−0.309 − 0.951i)8-s − 1.95i·10-s + (3.11 − 1.14i)11-s + (−3.07 − 4.23i)13-s + (−0.951 + 0.309i)14-s + (−0.809 − 0.587i)16-s + (−5.56 − 4.04i)17-s + (−3.10 + 1.00i)19-s + (−1.14 − 1.58i)20-s + (1.84 − 2.75i)22-s + 4.67i·23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (0.513 − 0.707i)5-s + (−0.359 − 0.116i)7-s + (−0.109 − 0.336i)8-s − 0.618i·10-s + (0.938 − 0.346i)11-s + (−0.853 − 1.17i)13-s + (−0.254 + 0.0825i)14-s + (−0.202 − 0.146i)16-s + (−1.35 − 0.981i)17-s + (−0.712 + 0.231i)19-s + (−0.256 − 0.353i)20-s + (0.392 − 0.588i)22-s + 0.974i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.748 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.748 + 0.662i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.748 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.997915462\)
\(L(\frac12)\) \(\approx\) \(1.997915462\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-3.11 + 1.14i)T \)
good5 \( 1 + (-1.14 + 1.58i)T + (-1.54 - 4.75i)T^{2} \)
13 \( 1 + (3.07 + 4.23i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.56 + 4.04i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (3.10 - 1.00i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.67iT - 23T^{2} \)
29 \( 1 + (1.03 - 3.18i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.95 + 5.04i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.00 + 6.16i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.0804 - 0.247i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.320iT - 43T^{2} \)
47 \( 1 + (-2.31 + 0.753i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.65 + 3.65i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.57 + 0.513i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.29 + 4.54i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 1.70T + 67T^{2} \)
71 \( 1 + (0.800 - 1.10i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.13 - 0.693i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.06 + 9.72i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.89 + 1.37i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.82iT - 89T^{2} \)
97 \( 1 + (-11.6 + 8.49i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.392929038769661777866286432960, −8.715300205792780673772018247487, −7.55123294061310461645862316352, −6.62926133587795616280433671430, −5.78635543648648984680490726370, −5.01213317455208070192300361128, −4.16663386478894989350708682916, −3.07688763168455806580853680333, −2.02356590369524051047013701177, −0.63581480158872038851666847118, 1.99312104019782985176279974318, 2.77225995555875205250995830241, 4.22013549859164570348416699434, 4.57496018341154887464455770749, 6.10319992336352899267163307627, 6.61370077340308348893381946533, 6.94166026512508330958729839288, 8.323465798688277417365868859815, 9.010171497391253508302265985033, 9.885244766743479314079702058824

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