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Properties

Label 2-1110-185.117-c1-0-30
Degree $2$
Conductor $1110$
Sign $-0.188 + 0.981i$
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 + (−0.825 − 2.07i)5-s + (−0.707 − 0.707i)6-s + (1.45 + 1.45i)7-s + 8-s + 1.00i·9-s + (−0.825 − 2.07i)10-s − 3.60i·11-s + (−0.707 − 0.707i)12-s − 0.566·13-s + (1.45 + 1.45i)14-s + (−0.886 + 2.05i)15-s + 16-s − 3.56i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (−0.368 − 0.929i)5-s + (−0.288 − 0.288i)6-s + (0.550 + 0.550i)7-s + 0.353·8-s + 0.333i·9-s + (−0.260 − 0.657i)10-s − 1.08i·11-s + (−0.204 − 0.204i)12-s − 0.157·13-s + (0.389 + 0.389i)14-s + (−0.228 + 0.530i)15-s + 0.250·16-s − 0.864i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.188 + 0.981i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.188 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.947424157\)
\(L(\frac12)\) \(\approx\) \(1.947424157\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.825 + 2.07i)T \)
37 \( 1 + (-5.24 + 3.08i)T \)
good7 \( 1 + (-1.45 - 1.45i)T + 7iT^{2} \)
11 \( 1 + 3.60iT - 11T^{2} \)
13 \( 1 + 0.566T + 13T^{2} \)
17 \( 1 + 3.56iT - 17T^{2} \)
19 \( 1 + (-2.60 + 2.60i)T - 19iT^{2} \)
23 \( 1 + 2.32T + 23T^{2} \)
29 \( 1 + (0.784 + 0.784i)T + 29iT^{2} \)
31 \( 1 + (1.50 - 1.50i)T - 31iT^{2} \)
41 \( 1 + 8.53iT - 41T^{2} \)
43 \( 1 - 1.13T + 43T^{2} \)
47 \( 1 + (4.15 + 4.15i)T + 47iT^{2} \)
53 \( 1 + (4.48 - 4.48i)T - 53iT^{2} \)
59 \( 1 + (-5.54 + 5.54i)T - 59iT^{2} \)
61 \( 1 + (5.08 - 5.08i)T - 61iT^{2} \)
67 \( 1 + (-1.64 + 1.64i)T - 67iT^{2} \)
71 \( 1 + 3.57T + 71T^{2} \)
73 \( 1 + (-0.352 - 0.352i)T + 73iT^{2} \)
79 \( 1 + (0.260 - 0.260i)T - 79iT^{2} \)
83 \( 1 + (-2.52 + 2.52i)T - 83iT^{2} \)
89 \( 1 + (-5.23 - 5.23i)T + 89iT^{2} \)
97 \( 1 - 11.4iT - 97T^{2} \)
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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

−9.482395629179865659568252024075, −8.680855001667353439972838049742, −7.890679209783084234579113379996, −7.09877657240653063345609913234, −5.90720412988422948048156456725, −5.33316664982762556385435805356, −4.60622241727508322080414113151, −3.42189367401115060081087583400, −2.14257511365418134023854749824, −0.72706671580474012632284984997, 1.72668781668772680385927822949, 3.09539599051274830965033991593, 4.10017283951181256284025463828, 4.66871081050153743935713490384, 5.84068782537212388425928465667, 6.61391421586655806322277975480, 7.51612255780509437793541229173, 8.057602255744480301644205529082, 9.664050590140544728866764404467, 10.19413285756898141463016601197

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