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Properties

Label 2-1110-185.174-c1-0-29
Degree $2$
Conductor $1110$
Sign $0.958 + 0.285i$
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  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (2.23 + 0.133i)5-s + 0.999·6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (1.99 − i)10-s + 4·11-s + (0.866 − 0.499i)12-s + (2.59 + 1.5i)13-s − 0.999·14-s + (1.86 + 1.23i)15-s + (−0.5 − 0.866i)16-s + (−5.19 + 3i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (0.998 + 0.0599i)5-s + 0.408·6-s + (−0.327 − 0.188i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.632 − 0.316i)10-s + 1.20·11-s + (0.249 − 0.144i)12-s + (0.720 + 0.416i)13-s − 0.267·14-s + (0.481 + 0.318i)15-s + (−0.125 − 0.216i)16-s + (−1.26 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.958 + 0.285i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.958 + 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.321671091\)
\(L(\frac12)\) \(\approx\) \(3.321671091\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-2.23 - 0.133i)T \)
37 \( 1 + (6.06 + 0.5i)T \)
good7 \( 1 + (0.866 + 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + (-2.59 - 1.5i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.19 - 3i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
41 \( 1 + (-4 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + (-3.46 + 2i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.92 - 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 16iT - 73T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (14.7 - 8.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2 + 3.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−9.840064793648448531732818624111, −9.023698738487869361382676493034, −8.558944044702380179922315792510, −6.86161109838808018166750636445, −6.49048557871274682785351128969, −5.51856044178916352168380622289, −4.30583963995840568401795585800, −3.71099829979146312959561252384, −2.44058524680606392032232969073, −1.51261396364571870057011780790, 1.48793541248492360513160215024, 2.65156489108211342245910221244, 3.62768767910899127005893585977, 4.72917163355928303756984101527, 5.77988465528278070469430885902, 6.53297331472064380813366707199, 7.05631788231048121868682064448, 8.325949989328414860830321789498, 9.076230820918730932770792876265, 9.568579287724510814297961603617

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