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Properties

Label 1-3744-3744.1195-r1-0-0
Degree $1$
Conductor $3744$
Sign $0.518 - 0.854i$
Analytic cond. $402.348$
Root an. cond. $402.348$
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  + (−0.965 + 0.258i)5-s + (0.866 − 0.5i)7-s + (0.258 − 0.965i)11-s − 17-s + (−0.707 + 0.707i)19-s + (−0.866 − 0.5i)23-s + (0.866 − 0.5i)25-s + (−0.965 − 0.258i)29-s + (−0.5 + 0.866i)31-s + (−0.707 + 0.707i)35-s + (0.707 + 0.707i)37-s + (−0.866 − 0.5i)41-s + (−0.258 + 0.965i)43-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)5-s + (0.866 − 0.5i)7-s + (0.258 − 0.965i)11-s − 17-s + (−0.707 + 0.707i)19-s + (−0.866 − 0.5i)23-s + (0.866 − 0.5i)25-s + (−0.965 − 0.258i)29-s + (−0.5 + 0.866i)31-s + (−0.707 + 0.707i)35-s + (0.707 + 0.707i)37-s + (−0.866 − 0.5i)41-s + (−0.258 + 0.965i)43-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.518 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.518 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.518 - 0.854i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ 0.518 - 0.854i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024049792 - 0.5764338546i\)
\(L(\frac12)\) \(\approx\) \(1.024049792 - 0.5764338546i\)
\(L(1)\) \(\approx\) \(0.8589852836 - 0.04900712899i\)
\(L(1)\) \(\approx\) \(0.8589852836 - 0.04900712899i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.258 - 0.965i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.965 - 0.258i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (-0.258 + 0.965i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.965 - 0.258i)T \)
61 \( 1 + (-0.965 - 0.258i)T \)
67 \( 1 + (0.258 + 0.965i)T \)
71 \( 1 - iT \)
73 \( 1 - iT \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.965 + 0.258i)T \)
89 \( 1 - iT \)
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

−18.3275743498034089679691617202, −18.06536674546118793924318328591, −17.14192440018966683065607872991, −16.58769781256738032219293205196, −15.57113233010073833587903462539, −15.10865673156162786882726727192, −14.81416270385900421288038953917, −13.69192240371403805363784697298, −12.95799201598574713420667610870, −12.25217478963680908712424308583, −11.588697626185241363934521981312, −11.16461276359593991883999887294, −10.308083548100745514847881980276, −9.21406320535720877721319046936, −8.81958662119861715557430567170, −7.95740812755552820699270987452, −7.38858102477180495677393885657, −6.659210014269666827398899222695, −5.6122488206938515212484478191, −4.82063553651934645717036215747, −4.25095845619998040614344779556, −3.582702145325150323045187018078, −2.20840084312393772423303981934, −1.8860448700755859449048890693, −0.53381221528277948113287137949, 0.29846442461431690013693492963, 1.265168762107562819867974977570, 2.212501386183366594501866690154, 3.22942861816886335796807804856, 4.07683302576434719684159576651, 4.40902396493304497895375812874, 5.49319666765501727074640453952, 6.36525897027661388276885640841, 7.06413456688471918752575528181, 7.918903336034868610854840233, 8.36225925280044218146173859467, 9.0129178637537346138611459316, 10.2333886813481006405399502347, 10.80772318651047136949473462255, 11.38596981538173696604781302068, 11.907423560987984937169407828309, 12.8244081937497506871025456822, 13.58221001864268257196744919796, 14.41133257290162702282791137340, 14.752549538951879741477145400852, 15.63056298093374826566319400316, 16.291878388417241972831073906734, 16.90492355217878183014132903640, 17.63354288622271825536323173447, 18.53143162666442968482410101350

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