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Properties

Label 1-1368-1368.277-r0-0-0
Degree $1$
Conductor $1368$
Sign $-0.884 + 0.466i$
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  − 5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)23-s + 25-s − 29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)35-s − 37-s + 41-s + (0.5 − 0.866i)43-s + 47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  − 5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)23-s + 25-s − 29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)35-s − 37-s + 41-s + (0.5 − 0.866i)43-s + 47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.884 + 0.466i$
Analytic conductor: \(6.35296\)
Root analytic conductor: \(6.35296\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1368,\ (0:\ ),\ -0.884 + 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03984451536 + 0.1609776608i\)
\(L(\frac12)\) \(\approx\) \(0.03984451536 + 0.1609776608i\)
\(L(1)\) \(\approx\) \(0.7049365465 + 0.007649069320i\)
\(L(1)\) \(\approx\) \(0.7049365465 + 0.007649069320i\)

Euler product

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

−20.33805667518492519175175881439, −19.72409165614498160144707594897, −19.02573018083179425161857826315, −18.50695025734717086749451818072, −17.50610031199499205074044110197, −16.62417124264156023338579505020, −15.79409728748277046124036955050, −15.38695775774420850109017312688, −14.62648274149219163854496506426, −13.52250929588814411069436377986, −12.76320852048508116816180145722, −12.08793790738842056160511398306, −11.22981696745396955254345548439, −10.71111698680696081495595764504, −9.43367827384889587733183832941, −8.76228417564756142400837511025, −8.05455990446194709064833188934, −7.22893683050576882096862232929, −5.988303913087701330760467717861, −5.7109543970742828109284468010, −4.244629522895828105553767374694, −3.565973016925054631619478721561, −2.80065457656948551467383627689, −1.47169186010714376687113088361, −0.06914035860307639177162394083, 1.24628893829255376288044211894, 2.46484887518543962642159051454, 3.754903656295033969463565350919, 4.09957110922230538574905173277, 5.014951545129585210559159397367, 6.43562399923214310756256450350, 7.05943683757864975708222302759, 7.60154079561616636434880795889, 8.81405672302420407181265568016, 9.36715373490099164950480141664, 10.488247164316427610876717340418, 11.08946907758827643671924001413, 12.03147461101139509179388992783, 12.55575540530996498068953421707, 13.628307167288036318545294764, 14.2629874831633082972242148158, 15.1353563473377018978690539665, 16.03137891347336253093132841769, 16.434214616368626745680389209209, 17.311713308045400422631982364739, 18.24868953591588535727616866832, 19.00461781016286862937554271926, 19.75028595751560633439619472284, 20.32347212938602692339557038323, 20.88932961866355687213160034854

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