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Properties

Label 2-1110-185.43-c1-0-20
Degree $2$
Conductor $1110$
Sign $-0.259 + 0.965i$
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 + (−0.707 + 0.707i)3-s − 4-s + (0.248 − 2.22i)5-s + (0.707 + 0.707i)6-s + (−2.88 + 2.88i)7-s + i·8-s − 1.00i·9-s + (−2.22 − 0.248i)10-s + 4.93i·11-s + (0.707 − 0.707i)12-s − 1.83i·13-s + (2.88 + 2.88i)14-s + (1.39 + 1.74i)15-s + 16-s + 2.74·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (0.111 − 0.993i)5-s + (0.288 + 0.288i)6-s + (−1.09 + 1.09i)7-s + 0.353i·8-s − 0.333i·9-s + (−0.702 − 0.0787i)10-s + 1.48i·11-s + (0.204 − 0.204i)12-s − 0.507i·13-s + (0.772 + 0.772i)14-s + (0.360 + 0.451i)15-s + 0.250·16-s + 0.665·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.259 + 0.965i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.259 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9452838136\)
\(L(\frac12)\) \(\approx\) \(0.9452838136\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.248 + 2.22i)T \)
37 \( 1 + (2.83 + 5.38i)T \)
good7 \( 1 + (2.88 - 2.88i)T - 7iT^{2} \)
11 \( 1 - 4.93iT - 11T^{2} \)
13 \( 1 + 1.83iT - 13T^{2} \)
17 \( 1 - 2.74T + 17T^{2} \)
19 \( 1 + (-5.44 + 5.44i)T - 19iT^{2} \)
23 \( 1 + 8.34iT - 23T^{2} \)
29 \( 1 + (0.469 + 0.469i)T + 29iT^{2} \)
31 \( 1 + (-2.48 + 2.48i)T - 31iT^{2} \)
41 \( 1 - 4.25iT - 41T^{2} \)
43 \( 1 + 5.61iT - 43T^{2} \)
47 \( 1 + (0.544 - 0.544i)T - 47iT^{2} \)
53 \( 1 + (7.02 + 7.02i)T + 53iT^{2} \)
59 \( 1 + (7.44 - 7.44i)T - 59iT^{2} \)
61 \( 1 + (-1.23 + 1.23i)T - 61iT^{2} \)
67 \( 1 + (-5.74 - 5.74i)T + 67iT^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + (-5.94 + 5.94i)T - 73iT^{2} \)
79 \( 1 + (-4.75 + 4.75i)T - 79iT^{2} \)
83 \( 1 + (0.579 + 0.579i)T + 83iT^{2} \)
89 \( 1 + (2.52 + 2.52i)T + 89iT^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−9.645993681312722392382761760997, −9.168315784225813299037292117673, −8.270183591566252791636253361587, −7.03928071172846940789868056981, −5.93213709353133335253573407864, −5.12628179917803986944446605919, −4.48923199762689555002577285421, −3.20031973279770710626741857506, −2.19485965189074642782861569257, −0.51657882169733211109330037426, 1.15475483053419401810256804364, 3.35297917971319169032812934265, 3.56665480564363057372257731074, 5.32444032328852476498475020202, 6.11714120179524610473322218392, 6.64978872780147389718921363210, 7.50232468576200309295004837278, 8.015577655415576354821511361111, 9.474429839618292374676061138413, 9.950328934575585089774600476957

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