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Properties

Label 2-1110-111.68-c1-0-20
Degree $2$
Conductor $1110$
Sign $0.423 + 0.906i$
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  + (−0.707 + 0.707i)2-s + (−0.912 − 1.47i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (1.68 + 0.396i)6-s + 0.363·7-s + (0.707 + 0.707i)8-s + (−1.33 + 2.68i)9-s + 1.00·10-s + 1.23·11-s + (−1.47 + 0.912i)12-s + (2.54 − 2.54i)13-s + (−0.257 + 0.257i)14-s + (−0.396 + 1.68i)15-s − 1.00·16-s + (2.79 + 2.79i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.526 − 0.850i)3-s − 0.500i·4-s + (−0.316 − 0.316i)5-s + (0.688 + 0.161i)6-s + 0.137·7-s + (0.250 + 0.250i)8-s + (−0.445 + 0.895i)9-s + 0.316·10-s + 0.372·11-s + (−0.425 + 0.263i)12-s + (0.706 − 0.706i)13-s + (−0.0687 + 0.0687i)14-s + (−0.102 + 0.435i)15-s − 0.250·16-s + (0.677 + 0.677i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.423 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.423 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.423 + 0.906i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.423 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9464882451\)
\(L(\frac12)\) \(\approx\) \(0.9464882451\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.912 + 1.47i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-5.07 + 3.35i)T \)
good7 \( 1 - 0.363T + 7T^{2} \)
11 \( 1 - 1.23T + 11T^{2} \)
13 \( 1 + (-2.54 + 2.54i)T - 13iT^{2} \)
17 \( 1 + (-2.79 - 2.79i)T + 17iT^{2} \)
19 \( 1 + (-1.40 + 1.40i)T - 19iT^{2} \)
23 \( 1 + (-0.456 - 0.456i)T + 23iT^{2} \)
29 \( 1 + (1.61 - 1.61i)T - 29iT^{2} \)
31 \( 1 + (-0.163 - 0.163i)T + 31iT^{2} \)
41 \( 1 + 0.545T + 41T^{2} \)
43 \( 1 + (-4.75 + 4.75i)T - 43iT^{2} \)
47 \( 1 - 1.60iT - 47T^{2} \)
53 \( 1 + 8.02iT - 53T^{2} \)
59 \( 1 + (6.45 + 6.45i)T + 59iT^{2} \)
61 \( 1 + (-2.60 - 2.60i)T + 61iT^{2} \)
67 \( 1 + 4.94iT - 67T^{2} \)
71 \( 1 + 14.8iT - 71T^{2} \)
73 \( 1 + 1.60iT - 73T^{2} \)
79 \( 1 + (-2.32 + 2.32i)T - 79iT^{2} \)
83 \( 1 + 8.43iT - 83T^{2} \)
89 \( 1 + (0.550 - 0.550i)T - 89iT^{2} \)
97 \( 1 + (0.870 - 0.870i)T - 97iT^{2} \)
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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

−9.560555403850519260164994000559, −8.618473336494703785237468306188, −7.948985074117520849364883345326, −7.34531307155126148475194076607, −6.33107753865939908603598270917, −5.70429334066448769975466659764, −4.79216997100682226820425946150, −3.39422598366043937630668341532, −1.75971023383099839064257449961, −0.66590441805080327831009512988, 1.12473024652248367994678504649, 2.84601871641749873191089022985, 3.80234359370962435763584697460, 4.55908419053695500121393294146, 5.74435751354832324626543358354, 6.63019378595874339302590619605, 7.63151242985010228578722772472, 8.583589345674278969523287402227, 9.410615613781509018470487532395, 9.937089297783697200091991017246

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