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Properties

Label 2-1110-185.142-c1-0-1
Degree $2$
Conductor $1110$
Sign $-0.772 - 0.635i$
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 + (0.262 + 2.22i)5-s + (0.707 − 0.707i)6-s + (−2.69 − 2.69i)7-s + i·8-s + 1.00i·9-s + (2.22 − 0.262i)10-s + 3.85i·11-s + (−0.707 − 0.707i)12-s − 1.07i·13-s + (−2.69 + 2.69i)14-s + (−1.38 + 1.75i)15-s + 16-s − 3.12·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (0.117 + 0.993i)5-s + (0.288 − 0.288i)6-s + (−1.01 − 1.01i)7-s + 0.353i·8-s + 0.333i·9-s + (0.702 − 0.0829i)10-s + 1.16i·11-s + (−0.204 − 0.204i)12-s − 0.297i·13-s + (−0.720 + 0.720i)14-s + (−0.357 + 0.453i)15-s + 0.250·16-s − 0.758·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.772 - 0.635i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.772 - 0.635i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3701324021\)
\(L(\frac12)\) \(\approx\) \(0.3701324021\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.262 - 2.22i)T \)
37 \( 1 + (-0.778 - 6.03i)T \)
good7 \( 1 + (2.69 + 2.69i)T + 7iT^{2} \)
11 \( 1 - 3.85iT - 11T^{2} \)
13 \( 1 + 1.07iT - 13T^{2} \)
17 \( 1 + 3.12T + 17T^{2} \)
19 \( 1 + (4.87 + 4.87i)T + 19iT^{2} \)
23 \( 1 - 6.72iT - 23T^{2} \)
29 \( 1 + (-0.240 + 0.240i)T - 29iT^{2} \)
31 \( 1 + (7.29 + 7.29i)T + 31iT^{2} \)
41 \( 1 + 9.24iT - 41T^{2} \)
43 \( 1 - 8.98iT - 43T^{2} \)
47 \( 1 + (3.13 + 3.13i)T + 47iT^{2} \)
53 \( 1 + (6.75 - 6.75i)T - 53iT^{2} \)
59 \( 1 + (1.03 + 1.03i)T + 59iT^{2} \)
61 \( 1 + (-9.33 - 9.33i)T + 61iT^{2} \)
67 \( 1 + (0.0813 - 0.0813i)T - 67iT^{2} \)
71 \( 1 + 9.73T + 71T^{2} \)
73 \( 1 + (3.77 + 3.77i)T + 73iT^{2} \)
79 \( 1 + (4.33 + 4.33i)T + 79iT^{2} \)
83 \( 1 + (5.06 - 5.06i)T - 83iT^{2} \)
89 \( 1 + (4.92 - 4.92i)T - 89iT^{2} \)
97 \( 1 + 2.13T + 97T^{2} \)
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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

−9.972374919195733905219130979726, −9.770041178560097546874325306899, −8.839155118614657757330094809404, −7.48151605940727781734946161880, −7.04995803302559495697817103547, −6.00199256118111196226766224838, −4.56606162679611113109736052678, −3.86002569267687668255941952362, −2.98031392164453838178859049819, −2.01703255640889768993495996328, 0.14302745806646174416413184252, 1.94372537872346630278942933752, 3.25612804857310862759635328267, 4.33165289998206069782107436611, 5.53275476468840521885165533556, 6.16125743338108523610077589318, 6.82352622278492950392669747403, 8.205420669428558645851479202860, 8.652414667720106577302782232938, 9.083568144043737478977859007556

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