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Properties

Label 2-1110-185.142-c1-0-32
Degree $2$
Conductor $1110$
Sign $-0.974 + 0.222i$
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 + (2.05 + 0.891i)5-s + (−0.707 + 0.707i)6-s + (−1.56 − 1.56i)7-s + i·8-s + 1.00i·9-s + (0.891 − 2.05i)10-s − 3.92i·11-s + (0.707 + 0.707i)12-s + 0.319i·13-s + (−1.56 + 1.56i)14-s + (−0.819 − 2.08i)15-s + 16-s + 1.25·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (0.917 + 0.398i)5-s + (−0.288 + 0.288i)6-s + (−0.591 − 0.591i)7-s + 0.353i·8-s + 0.333i·9-s + (0.282 − 0.648i)10-s − 1.18i·11-s + (0.204 + 0.204i)12-s + 0.0885i·13-s + (−0.418 + 0.418i)14-s + (−0.211 − 0.537i)15-s + 0.250·16-s + 0.304·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.974 + 0.222i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.974 + 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.039897715\)
\(L(\frac12)\) \(\approx\) \(1.039897715\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.05 - 0.891i)T \)
37 \( 1 + (-0.0100 + 6.08i)T \)
good7 \( 1 + (1.56 + 1.56i)T + 7iT^{2} \)
11 \( 1 + 3.92iT - 11T^{2} \)
13 \( 1 - 0.319iT - 13T^{2} \)
17 \( 1 - 1.25T + 17T^{2} \)
19 \( 1 + (2.17 + 2.17i)T + 19iT^{2} \)
23 \( 1 - 1.15iT - 23T^{2} \)
29 \( 1 + (-1.22 + 1.22i)T - 29iT^{2} \)
31 \( 1 + (6.78 + 6.78i)T + 31iT^{2} \)
41 \( 1 + 3.92iT - 41T^{2} \)
43 \( 1 - 1.87iT - 43T^{2} \)
47 \( 1 + (4.88 + 4.88i)T + 47iT^{2} \)
53 \( 1 + (3.03 - 3.03i)T - 53iT^{2} \)
59 \( 1 + (4.70 + 4.70i)T + 59iT^{2} \)
61 \( 1 + (5.56 + 5.56i)T + 61iT^{2} \)
67 \( 1 + (6.88 - 6.88i)T - 67iT^{2} \)
71 \( 1 + 3.29T + 71T^{2} \)
73 \( 1 + (-8.53 - 8.53i)T + 73iT^{2} \)
79 \( 1 + (-3.59 - 3.59i)T + 79iT^{2} \)
83 \( 1 + (1.13 - 1.13i)T - 83iT^{2} \)
89 \( 1 + (1.33 - 1.33i)T - 89iT^{2} \)
97 \( 1 - 4.01T + 97T^{2} \)
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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

−9.597117257676220963967875911477, −8.908796939744377554103499450438, −7.78308991951030774666217833546, −6.80675493499844966748002892588, −6.02888379050046764690095448331, −5.31509736677735945732380111685, −3.93841277591278239286473546813, −2.99272026340767367794082738993, −1.87723683859071734073015142074, −0.47713989622159454876859585092, 1.66364095855178720391674128002, 3.12910842690529033409904915497, 4.53397288699849762950939953407, 5.15166860865391047193251242424, 6.07430972165039190893437264406, 6.59789128313644509884574874464, 7.68209483785444186980593663046, 8.797954060141284018044016989188, 9.354795707155813439088109657328, 10.04829122103855728976179671309

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