Properties

Label 2-1110-185.68-c1-0-9
Degree $2$
Conductor $1110$
Sign $0.0337 - 0.999i$
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  − 2-s + (−0.707 + 0.707i)3-s + 4-s + (1.63 + 1.52i)5-s + (0.707 − 0.707i)6-s + (1.84 − 1.84i)7-s − 8-s − 1.00i·9-s + (−1.63 − 1.52i)10-s + 3.01i·11-s + (−0.707 + 0.707i)12-s − 1.82·13-s + (−1.84 + 1.84i)14-s + (−2.23 + 0.0778i)15-s + 16-s + 5.42i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (0.731 + 0.682i)5-s + (0.288 − 0.288i)6-s + (0.696 − 0.696i)7-s − 0.353·8-s − 0.333i·9-s + (−0.517 − 0.482i)10-s + 0.907i·11-s + (−0.204 + 0.204i)12-s − 0.504·13-s + (−0.492 + 0.492i)14-s + (−0.577 + 0.0201i)15-s + 0.250·16-s + 1.31i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0337 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0337 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.0337 - 0.999i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.0337 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.131828794\)
\(L(\frac12)\) \(\approx\) \(1.131828794\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1.63 - 1.52i)T \)
37 \( 1 + (-2.27 + 5.64i)T \)
good7 \( 1 + (-1.84 + 1.84i)T - 7iT^{2} \)
11 \( 1 - 3.01iT - 11T^{2} \)
13 \( 1 + 1.82T + 13T^{2} \)
17 \( 1 - 5.42iT - 17T^{2} \)
19 \( 1 + (-0.509 - 0.509i)T + 19iT^{2} \)
23 \( 1 + 0.0216T + 23T^{2} \)
29 \( 1 + (-4.67 + 4.67i)T - 29iT^{2} \)
31 \( 1 + (-1.55 - 1.55i)T + 31iT^{2} \)
41 \( 1 - 6.35iT - 41T^{2} \)
43 \( 1 + 7.86T + 43T^{2} \)
47 \( 1 + (0.176 - 0.176i)T - 47iT^{2} \)
53 \( 1 + (-2.05 - 2.05i)T + 53iT^{2} \)
59 \( 1 + (-2.84 - 2.84i)T + 59iT^{2} \)
61 \( 1 + (-4.51 - 4.51i)T + 61iT^{2} \)
67 \( 1 + (3.93 + 3.93i)T + 67iT^{2} \)
71 \( 1 - 9.00T + 71T^{2} \)
73 \( 1 + (8.50 - 8.50i)T - 73iT^{2} \)
79 \( 1 + (-2.16 - 2.16i)T + 79iT^{2} \)
83 \( 1 + (-0.925 - 0.925i)T + 83iT^{2} \)
89 \( 1 + (1.03 - 1.03i)T - 89iT^{2} \)
97 \( 1 - 7.65iT - 97T^{2} \)
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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

−10.10843180964283949726515689328, −9.541156781664266057162596119327, −8.388825665576650101026446967273, −7.55963561534454658600691500499, −6.76551530391997523333826804416, −5.99048190649025927696165741460, −4.90611985865544569269026186168, −3.93083489980825639193090099544, −2.50185121800187366452543953949, −1.40408887349265738513629898455, 0.70614126998679070078507854497, 1.89815003938827984612924755446, 2.91768235320145381639166727808, 4.85696547041282814705718017420, 5.37135919818157553548468586695, 6.30487987476584033284403260407, 7.18190150735109646204849094798, 8.269473989619263425740413665962, 8.704758953784434850758703748451, 9.576813782368923865082931552406

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