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Properties

Label 2-1110-185.68-c1-0-5
Degree $2$
Conductor $1110$
Sign $0.229 - 0.973i$
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 + (1.21 − 1.87i)5-s + (0.707 − 0.707i)6-s + (−2.24 + 2.24i)7-s − 8-s − 1.00i·9-s + (−1.21 + 1.87i)10-s − 5.65i·11-s + (−0.707 + 0.707i)12-s − 4.81·13-s + (2.24 − 2.24i)14-s + (0.466 + 2.18i)15-s + 16-s + 7.58i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (0.543 − 0.839i)5-s + (0.288 − 0.288i)6-s + (−0.849 + 0.849i)7-s − 0.353·8-s − 0.333i·9-s + (−0.384 + 0.593i)10-s − 1.70i·11-s + (−0.204 + 0.204i)12-s − 1.33·13-s + (0.600 − 0.600i)14-s + (0.120 + 0.564i)15-s + 0.250·16-s + 1.83i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.229 - 0.973i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7252836472\)
\(L(\frac12)\) \(\approx\) \(0.7252836472\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.21 + 1.87i)T \)
37 \( 1 + (-4.42 - 4.17i)T \)
good7 \( 1 + (2.24 - 2.24i)T - 7iT^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 + 4.81T + 13T^{2} \)
17 \( 1 - 7.58iT - 17T^{2} \)
19 \( 1 + (-3.73 - 3.73i)T + 19iT^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + (3.26 - 3.26i)T - 29iT^{2} \)
31 \( 1 + (-1.98 - 1.98i)T + 31iT^{2} \)
41 \( 1 - 2.43iT - 41T^{2} \)
43 \( 1 + 3.34T + 43T^{2} \)
47 \( 1 + (-6.04 + 6.04i)T - 47iT^{2} \)
53 \( 1 + (-8.02 - 8.02i)T + 53iT^{2} \)
59 \( 1 + (-9.84 - 9.84i)T + 59iT^{2} \)
61 \( 1 + (1.11 + 1.11i)T + 61iT^{2} \)
67 \( 1 + (6.91 + 6.91i)T + 67iT^{2} \)
71 \( 1 + 6.12T + 71T^{2} \)
73 \( 1 + (-2.58 + 2.58i)T - 73iT^{2} \)
79 \( 1 + (-2.85 - 2.85i)T + 79iT^{2} \)
83 \( 1 + (-0.906 - 0.906i)T + 83iT^{2} \)
89 \( 1 + (10.1 - 10.1i)T - 89iT^{2} \)
97 \( 1 - 2.87iT - 97T^{2} \)
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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

−10.02831597925435419167112551538, −9.056736460077801466884068390898, −8.755072362411903169651326113557, −7.77287044643615894900259118385, −6.41653272848487616904408799088, −5.81700610005799874356245628517, −5.21544699169203962088393621746, −3.70158179522259872022575355809, −2.64842472072368153574063741978, −1.12646936350786516248238670344, 0.47969899090595512136776746106, 2.20921520589345023033451153257, 2.93495869840169836303336278620, 4.56345993188464450114737081074, 5.52600010425382766322753662774, 6.88300834674371926992132166026, 7.18318768028196319883742867701, 7.42595795160212394509504259550, 9.300051382740138435448682461314, 9.791039608409843029334632886274

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