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Properties

Label 1-1368-1368.349-r0-0-0
Degree $1$
Conductor $1368$
Sign $-0.612 + 0.790i$
Analytic cond. $6.35296$
Root an. cond. $6.35296$
Motivic weight $0$
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)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s − 13-s + (−0.5 + 0.866i)17-s + 23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.5 − 0.866i)35-s − 37-s + (−0.5 − 0.866i)41-s − 43-s + (−0.5 + 0.866i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s − 13-s + (−0.5 + 0.866i)17-s + 23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.5 − 0.866i)35-s − 37-s + (−0.5 − 0.866i)41-s − 43-s + (−0.5 + 0.866i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.612 + 0.790i$
Analytic conductor: \(6.35296\)
Root analytic conductor: \(6.35296\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1368,\ (0:\ ),\ -0.612 + 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4150665642 + 0.8473387740i\)
\(L(\frac12)\) \(\approx\) \(0.4150665642 + 0.8473387740i\)
\(L(1)\) \(\approx\) \(0.9091898446 + 0.2457937050i\)
\(L(1)\) \(\approx\) \(0.9091898446 + 0.2457937050i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.618038749342028243319601084891, −19.754790376355429377810747549713, −19.19840934933944915013802577161, −18.285049976514161347253445472437, −17.55624547502664490505151175816, −16.53682055558378979604963108492, −16.36226597438435790487962346465, −15.23308321066840489879837648071, −14.53613700648350162511514416057, −13.471675642251119248764713724474, −13.03885216623855719850737476175, −12.02363370075501039099106368823, −11.61696320176330061884399349205, −10.363235951869052809019744918269, −9.457828127766789560251082537548, −8.98452684570147418787656023069, −8.298119114757725920771690797624, −7.0197086426307664213330244346, −6.29527485312490391762387203551, −5.25923465441104693238470117594, −4.89118867078897393262037972227, −3.49756456121823600295447880172, −2.6229545594380529319154285961, −1.655197620421106802025928170744, −0.34720740553128091650662331113, 1.42175146593250448395857283947, 2.37217545786619454478635284329, 3.32875562121847939564108620683, 4.207436649967280686464307907, 5.149123947950982974317247485319, 6.32730419068525666869327972334, 6.95987050812091473897078758201, 7.42727942148654703822905279249, 8.73987300777365362871481059238, 9.69115749190262718433232680035, 10.237009671684132874883229280285, 10.86082427278957134193982270566, 11.92056175974047296522621784426, 12.76143075221031991814850490850, 13.51049859421836863903016933737, 14.32195962181258148287529379856, 14.92621778142091632111486961084, 15.65678734761183754152442293188, 16.90587325349939831751395561533, 17.2956662456813655724338919488, 17.90443241530019696165824412534, 19.08829695365267500011725602626, 19.49121295625580481376208577925, 20.27181332634753752112448663534, 21.1940201451709663040560477724

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