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Properties

Label 2-1386-77.10-c1-0-35
Degree $2$
Conductor $1386$
Sign $0.0318 + 0.999i$
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 + (2.13 − 1.23i)5-s + (0.941 − 2.47i)7-s − 0.999i·8-s + (1.23 − 2.13i)10-s + (2.32 − 2.36i)11-s + 1.32·13-s + (−0.420 − 2.61i)14-s + (−0.5 − 0.866i)16-s + (−2.23 + 3.87i)17-s + (2.21 + 3.83i)19-s − 2.46i·20-s + (0.834 − 3.20i)22-s + (−4.14 − 7.17i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.953 − 0.550i)5-s + (0.356 − 0.934i)7-s − 0.353i·8-s + (0.389 − 0.674i)10-s + (0.701 − 0.712i)11-s + 0.366·13-s + (−0.112 − 0.698i)14-s + (−0.125 − 0.216i)16-s + (−0.542 + 0.939i)17-s + (0.508 + 0.880i)19-s − 0.550i·20-s + (0.178 − 0.684i)22-s + (−0.864 − 1.49i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0318 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0318 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $0.0318 + 0.999i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 0.0318 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.022091719\)
\(L(\frac12)\) \(\approx\) \(3.022091719\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.941 + 2.47i)T \)
11 \( 1 + (-2.32 + 2.36i)T \)
good5 \( 1 + (-2.13 + 1.23i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 1.32T + 13T^{2} \)
17 \( 1 + (2.23 - 3.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.21 - 3.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.14 + 7.17i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.44iT - 29T^{2} \)
31 \( 1 + (-2.34 - 1.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.46 - 4.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.18T + 41T^{2} \)
43 \( 1 - 9.63iT - 43T^{2} \)
47 \( 1 + (-0.664 + 0.383i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.945 - 1.63i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.74 + 3.89i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.23 - 3.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.861 + 1.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + (-5.58 + 9.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.53 - 1.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + (-10.9 + 6.32i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.37iT - 97T^{2} \)
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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

−9.572105603965273572639552527672, −8.566152828240498083633876892050, −7.904710776573310829846816753755, −6.35575263989075519347566575830, −6.27670050397673687995568098929, −5.05336306061410169441110419740, −4.24082800361461041940343083213, −3.41157641110882033445249022857, −1.93652784940158120406273581973, −1.08999504810714478145271694951, 1.83797920058633158711204453235, 2.60800359739267545720802324337, 3.77991566094672720605020207430, 4.97429534133545404004708954608, 5.57677368226915528485371941184, 6.46210439829801649494137274428, 7.06571161121567680866805204456, 8.047574837969010579843255342689, 9.199536378978735444624325194413, 9.495429852180207508382401391538

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