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Properties

Label 2-1110-185.142-c1-0-35
Degree $2$
Conductor $1110$
Sign $-0.908 - 0.417i$
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 + (1.11 − 1.94i)5-s + (−0.707 + 0.707i)6-s + (0.636 + 0.636i)7-s + i·8-s + 1.00i·9-s + (−1.94 − 1.11i)10-s + 2.26i·11-s + (0.707 + 0.707i)12-s − 1.29i·13-s + (0.636 − 0.636i)14-s + (−2.15 + 0.586i)15-s + 16-s − 5.74·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (0.496 − 0.867i)5-s + (−0.288 + 0.288i)6-s + (0.240 + 0.240i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.613 − 0.351i)10-s + 0.684i·11-s + (0.204 + 0.204i)12-s − 0.358i·13-s + (0.170 − 0.170i)14-s + (−0.557 + 0.151i)15-s + 0.250·16-s − 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.908 - 0.417i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.908 - 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7637147887\)
\(L(\frac12)\) \(\approx\) \(0.7637147887\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 + 1.94i)T \)
37 \( 1 + (-5.19 - 3.16i)T \)
good7 \( 1 + (-0.636 - 0.636i)T + 7iT^{2} \)
11 \( 1 - 2.26iT - 11T^{2} \)
13 \( 1 + 1.29iT - 13T^{2} \)
17 \( 1 + 5.74T + 17T^{2} \)
19 \( 1 + (4.10 + 4.10i)T + 19iT^{2} \)
23 \( 1 + 8.53iT - 23T^{2} \)
29 \( 1 + (1.70 - 1.70i)T - 29iT^{2} \)
31 \( 1 + (5.81 + 5.81i)T + 31iT^{2} \)
41 \( 1 - 9.02iT - 41T^{2} \)
43 \( 1 + 0.853iT - 43T^{2} \)
47 \( 1 + (0.326 + 0.326i)T + 47iT^{2} \)
53 \( 1 + (6.79 - 6.79i)T - 53iT^{2} \)
59 \( 1 + (-3.51 - 3.51i)T + 59iT^{2} \)
61 \( 1 + (2.06 + 2.06i)T + 61iT^{2} \)
67 \( 1 + (-3.98 + 3.98i)T - 67iT^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 + (11.8 + 11.8i)T + 73iT^{2} \)
79 \( 1 + (-9.04 - 9.04i)T + 79iT^{2} \)
83 \( 1 + (-9.12 + 9.12i)T - 83iT^{2} \)
89 \( 1 + (6.32 - 6.32i)T - 89iT^{2} \)
97 \( 1 - 9.28T + 97T^{2} \)
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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

−9.314043929632035067649748881116, −8.761534837942006378271356909759, −7.928827970001401536072696498624, −6.70813490976889192236658971881, −5.93472278028792711676577530424, −4.70522425803776006751209081297, −4.47115907586272892984806832560, −2.55787752020733768046123466278, −1.82270449456122071460864907250, −0.33749020159285373491660657624, 1.87390911993904624268279592951, 3.44214753839720485336137234619, 4.26248141806560258373368655521, 5.47671297642807104334377889013, 6.06526612780173410656779240187, 6.88288256924893834540897404539, 7.64007068301927259803896103628, 8.763457002193468911372457247607, 9.407168299168708144897171418309, 10.33767505307435472664910302793

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