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Properties

Label 1-368-368.77-r0-0-0
Degree $1$
Conductor $368$
Sign $0.910 - 0.414i$
Analytic cond. $1.70898$
Root an. cond. $1.70898$
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  + (−0.755 − 0.654i)3-s + (0.989 + 0.142i)5-s + (−0.415 − 0.909i)7-s + (0.142 + 0.989i)9-s + (0.281 + 0.959i)11-s + (0.909 + 0.415i)13-s + (−0.654 − 0.755i)15-s + (0.841 − 0.540i)17-s + (−0.540 + 0.841i)19-s + (−0.281 + 0.959i)21-s + (0.959 + 0.281i)25-s + (0.540 − 0.841i)27-s + (0.540 + 0.841i)29-s + (−0.654 − 0.755i)31-s + (0.415 − 0.909i)33-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)3-s + (0.989 + 0.142i)5-s + (−0.415 − 0.909i)7-s + (0.142 + 0.989i)9-s + (0.281 + 0.959i)11-s + (0.909 + 0.415i)13-s + (−0.654 − 0.755i)15-s + (0.841 − 0.540i)17-s + (−0.540 + 0.841i)19-s + (−0.281 + 0.959i)21-s + (0.959 + 0.281i)25-s + (0.540 − 0.841i)27-s + (0.540 + 0.841i)29-s + (−0.654 − 0.755i)31-s + (0.415 − 0.909i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(1.70898\)
Root analytic conductor: \(1.70898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 368,\ (0:\ ),\ 0.910 - 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.193751291 - 0.2590118631i\)
\(L(\frac12)\) \(\approx\) \(1.193751291 - 0.2590118631i\)
\(L(1)\) \(\approx\) \(1.015859781 - 0.1663605244i\)
\(L(1)\) \(\approx\) \(1.015859781 - 0.1663605244i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.755 - 0.654i)T \)
5 \( 1 + (0.989 + 0.142i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
11 \( 1 + (0.281 + 0.959i)T \)
13 \( 1 + (0.909 + 0.415i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (-0.540 + 0.841i)T \)
29 \( 1 + (0.540 + 0.841i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + (-0.989 + 0.142i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 + T \)
53 \( 1 + (0.909 - 0.415i)T \)
59 \( 1 + (-0.909 - 0.415i)T \)
61 \( 1 + (0.755 - 0.654i)T \)
67 \( 1 + (0.281 - 0.959i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (0.989 - 0.142i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−24.7967010412535537896970154430, −23.75812118970021301732508542682, −22.826759272489448445169069671403, −21.83318794855933126954223876244, −21.512949025653454026985719734988, −20.70703853451983332370355607420, −19.30871089885728886395419762111, −18.37938719320017950635867734224, −17.56823009151106412303652811299, −16.72433027215483752175876921733, −15.93329199719624170760754360976, −15.09198802369340531458406460795, −13.94743185347206244293727725294, −12.924754575950449370037029032987, −12.064519022277541023218329944646, −10.95096835551389634454927120906, −10.23003620195033824359666760656, −9.14943959004118059719750843662, −8.56387011826508557695393815731, −6.602660912153015428003565758643, −5.85994594316633501970407193991, −5.337548414167224453676868043999, −3.849029927018281882972719552950, −2.72688173342709736890054689440, −1.08614117457175575492652438675, 1.1453638260178659412955029592, 2.08772695079734954235841807657, 3.730266847564758162476915870445, 5.0233480835417529953244609293, 6.0898551914162723298611031799, 6.80594422158828836379056921742, 7.65878776199945589682614825587, 9.17384569243865375917939634416, 10.22329686296302682044641749703, 10.80721985993721446494873801848, 12.11904547055211410884717587627, 12.875125835733035894984696431559, 13.77519779122973938851611050063, 14.42079443603022445566604285514, 16.04600821389959219463204363099, 16.83293393779245177823656809635, 17.456847567601877490785380593582, 18.35790160929993520463969133314, 19.06658205641835785169531337731, 20.31968957994857534301522433099, 21.06691874689230161024297473678, 22.21912817598177091246014584053, 22.96802199908747936355010464249, 23.5001937273281075802116070024, 24.6132133625317575034886801438

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