Properties

Label 2-1110-185.68-c1-0-16
Degree $2$
Conductor $1110$
Sign $0.560 + 0.828i$
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 + (−0.530 + 2.17i)5-s + (0.707 − 0.707i)6-s + (−1.93 + 1.93i)7-s − 8-s − 1.00i·9-s + (0.530 − 2.17i)10-s − 2.54i·11-s + (−0.707 + 0.707i)12-s − 2.64·13-s + (1.93 − 1.93i)14-s + (−1.16 − 1.91i)15-s + 16-s − 3.60i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (−0.237 + 0.971i)5-s + (0.288 − 0.288i)6-s + (−0.730 + 0.730i)7-s − 0.353·8-s − 0.333i·9-s + (0.167 − 0.686i)10-s − 0.766i·11-s + (−0.204 + 0.204i)12-s − 0.732·13-s + (0.516 − 0.516i)14-s + (−0.299 − 0.493i)15-s + 0.250·16-s − 0.874i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.560 + 0.828i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.560 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3987078212\)
\(L(\frac12)\) \(\approx\) \(0.3987078212\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.530 - 2.17i)T \)
37 \( 1 + (-5.87 - 1.59i)T \)
good7 \( 1 + (1.93 - 1.93i)T - 7iT^{2} \)
11 \( 1 + 2.54iT - 11T^{2} \)
13 \( 1 + 2.64T + 13T^{2} \)
17 \( 1 + 3.60iT - 17T^{2} \)
19 \( 1 + (2.47 + 2.47i)T + 19iT^{2} \)
23 \( 1 + 4.49T + 23T^{2} \)
29 \( 1 + (-4.58 + 4.58i)T - 29iT^{2} \)
31 \( 1 + (-7.33 - 7.33i)T + 31iT^{2} \)
41 \( 1 + 4.31iT - 41T^{2} \)
43 \( 1 + 3.12T + 43T^{2} \)
47 \( 1 + (-4.51 + 4.51i)T - 47iT^{2} \)
53 \( 1 + (0.301 + 0.301i)T + 53iT^{2} \)
59 \( 1 + (0.813 + 0.813i)T + 59iT^{2} \)
61 \( 1 + (0.867 + 0.867i)T + 61iT^{2} \)
67 \( 1 + (-5.77 - 5.77i)T + 67iT^{2} \)
71 \( 1 + 2.30T + 71T^{2} \)
73 \( 1 + (3.70 - 3.70i)T - 73iT^{2} \)
79 \( 1 + (4.70 + 4.70i)T + 79iT^{2} \)
83 \( 1 + (0.524 + 0.524i)T + 83iT^{2} \)
89 \( 1 + (-6.69 + 6.69i)T - 89iT^{2} \)
97 \( 1 + 17.0iT - 97T^{2} \)
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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

−9.968385763427571268907453021151, −8.969635768390180860921478735568, −8.199716678053942195398937419630, −7.11818330580808154535077994613, −6.43228125210117704427710305070, −5.74590789340708953030920407493, −4.46345891479709751284429210814, −3.11026502119964757601590423709, −2.51143749145169355432869945298, −0.27387868310003873803496342688, 1.04496924522144556308978499585, 2.28630187303525833410070002051, 3.91561686405404642229820190652, 4.73933193201284209080512778474, 6.01424771435013194105998691234, 6.66452406989692662515313681850, 7.73520459104895045855928573608, 8.105066570380167218124144292199, 9.255067175434566753976156297279, 9.992226428734148444620823584522

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