Properties

Label 2-1110-37.36-c1-0-21
Degree $2$
Conductor $1110$
Sign $0.806 - 0.590i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 3-s − 4-s + i·5-s + i·6-s + 2.75·7-s i·8-s + 9-s − 10-s + 4.75·11-s − 12-s − 6.75i·13-s + 2.75i·14-s + i·15-s + 16-s − 7.06i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s + 1.03·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 1.43·11-s − 0.288·12-s − 1.87i·13-s + 0.735i·14-s + 0.258i·15-s + 0.250·16-s − 1.71i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.806 - 0.590i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.806 - 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.277291278\)
\(L(\frac12)\) \(\approx\) \(2.277291278\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - T \)
5 \( 1 - iT \)
37 \( 1 + (4.90 - 3.59i)T \)
good7 \( 1 - 2.75T + 7T^{2} \)
11 \( 1 - 4.75T + 11T^{2} \)
13 \( 1 + 6.75iT - 13T^{2} \)
17 \( 1 + 7.06iT - 17T^{2} \)
19 \( 1 - 6.75iT - 19T^{2} \)
23 \( 1 + 5.06iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 0.315iT - 31T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 9.81iT - 43T^{2} \)
47 \( 1 - 3.50T + 47T^{2} \)
53 \( 1 - 4.75T + 53T^{2} \)
59 \( 1 - 11.8iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 - 15.0T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 6.43T + 73T^{2} \)
79 \( 1 - 8.31iT - 79T^{2} \)
83 \( 1 - 4.43T + 83T^{2} \)
89 \( 1 + 6.25iT - 89T^{2} \)
97 \( 1 - 11.5iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.857507936389302759441045408944, −8.888890386349630418019849483327, −8.126365975647094721634026138577, −7.60268433864957250425172581591, −6.69234505785433429918150981027, −5.72927241026616397126176919862, −4.80265329687410360886994779318, −3.79409821659877342865488685514, −2.74710474826647008075535668407, −1.16310788075165819611705531178, 1.48933829900666965542041427950, 1.97403564587966968325803893320, 3.73213394971237129634747897045, 4.21670720821100285176876464842, 5.15865534989914161451649026893, 6.51134889635301615228005387156, 7.34639350852534140463883131247, 8.635275026505422492192058799438, 8.845323486298143412218666662312, 9.551914491491157003672155178748

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