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Properties

Label 1-368-368.283-r0-0-0
Degree $1$
Conductor $368$
Sign $0.271 + 0.962i$
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  + (0.909 + 0.415i)3-s + (0.755 + 0.654i)5-s + (−0.841 + 0.540i)7-s + (0.654 + 0.755i)9-s + (−0.989 − 0.142i)11-s + (0.540 − 0.841i)13-s + (0.415 + 0.909i)15-s + (0.959 + 0.281i)17-s + (0.281 + 0.959i)19-s + (−0.989 + 0.142i)21-s + (0.142 + 0.989i)25-s + (0.281 + 0.959i)27-s + (−0.281 + 0.959i)29-s + (−0.415 − 0.909i)31-s + (−0.841 − 0.540i)33-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)3-s + (0.755 + 0.654i)5-s + (−0.841 + 0.540i)7-s + (0.654 + 0.755i)9-s + (−0.989 − 0.142i)11-s + (0.540 − 0.841i)13-s + (0.415 + 0.909i)15-s + (0.959 + 0.281i)17-s + (0.281 + 0.959i)19-s + (−0.989 + 0.142i)21-s + (0.142 + 0.989i)25-s + (0.281 + 0.959i)27-s + (−0.281 + 0.959i)29-s + (−0.415 − 0.909i)31-s + (−0.841 − 0.540i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.271 + 0.962i$
Analytic conductor: \(1.70898\)
Root analytic conductor: \(1.70898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 368,\ (0:\ ),\ 0.271 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.432742591 + 1.084546504i\)
\(L(\frac12)\) \(\approx\) \(1.432742591 + 1.084546504i\)
\(L(1)\) \(\approx\) \(1.355552530 + 0.5045579235i\)
\(L(1)\) \(\approx\) \(1.355552530 + 0.5045579235i\)

Euler product

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

−24.5181569066563768702163748038, −23.75236563319409696673840802445, −22.938740728807950322604856715662, −21.46225046954759073140811486272, −20.96411885870039746731168245954, −20.12253998775128655965720612280, −19.26833729514771729327023006696, −18.40685845255355619490574876584, −17.49935190156544044011887595380, −16.332322912499711748382106457807, −15.73007830650519008169663326318, −14.33110830343255933195134642520, −13.585061536392380236822753909533, −13.0512964790565773830554415108, −12.16763221147980784081511601227, −10.5858419500925938537296174937, −9.586228794735260906294501644141, −9.04382341876854060781787063310, −7.84646312111839403146328688625, −6.93470694598074144248539352364, −5.866738039425671371156033563553, −4.55950608655609211411882673531, −3.32709694034903130441772987648, −2.29184227694616175849137932217, −1.03525909316958739273604854784, 1.84356685031778090771460198355, 3.01320145047168925354883081330, 3.4748398551793464723583252172, 5.30934520563851132843119859571, 6.027859478289452991600028019029, 7.42136181859413129182857102861, 8.30935733071946600693001452501, 9.4541381713221276009979819620, 10.135119704419920516253779460120, 10.83316122399186386496090555752, 12.5746618555109011320922559165, 13.216483469527007667063796949418, 14.16635828508775210000444731849, 14.99591333674042193232690243935, 15.81323319350743476403109459706, 16.635502363199889444616535456158, 18.126020769052476993385985131034, 18.66717448903072250606615694038, 19.46761920209414258269319144335, 20.78970707686177712000163452804, 21.08292123656882590368249828037, 22.25707414799134646785351686281, 22.75315460327418211745545362632, 24.130579625717648395546587256770, 25.29132223663489656989815718117

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