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Properties

Label 2-1110-111.68-c1-0-24
Degree $2$
Conductor $1110$
Sign $0.999 - 0.0164i$
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 + (1.55 − 0.759i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (−0.564 + 1.63i)6-s + 1.92·7-s + (0.707 + 0.707i)8-s + (1.84 − 2.36i)9-s + 1.00·10-s + 5.18·11-s + (−0.759 − 1.55i)12-s + (−3.96 + 3.96i)13-s + (−1.36 + 1.36i)14-s + (−1.63 − 0.564i)15-s − 1.00·16-s + (3.83 + 3.83i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.898 − 0.438i)3-s − 0.500i·4-s + (−0.316 − 0.316i)5-s + (−0.230 + 0.668i)6-s + 0.727·7-s + (0.250 + 0.250i)8-s + (0.615 − 0.787i)9-s + 0.316·10-s + 1.56·11-s + (−0.219 − 0.449i)12-s + (−1.10 + 1.10i)13-s + (−0.363 + 0.363i)14-s + (−0.422 − 0.145i)15-s − 0.250·16-s + (0.930 + 0.930i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.999 - 0.0164i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.999 - 0.0164i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.923566741\)
\(L(\frac12)\) \(\approx\) \(1.923566741\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 + 0.759i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (5.86 - 1.59i)T \)
good7 \( 1 - 1.92T + 7T^{2} \)
11 \( 1 - 5.18T + 11T^{2} \)
13 \( 1 + (3.96 - 3.96i)T - 13iT^{2} \)
17 \( 1 + (-3.83 - 3.83i)T + 17iT^{2} \)
19 \( 1 + (-0.0989 + 0.0989i)T - 19iT^{2} \)
23 \( 1 + (-1.07 - 1.07i)T + 23iT^{2} \)
29 \( 1 + (-1.77 + 1.77i)T - 29iT^{2} \)
31 \( 1 + (4.45 + 4.45i)T + 31iT^{2} \)
41 \( 1 - 12.4T + 41T^{2} \)
43 \( 1 + (-7.21 + 7.21i)T - 43iT^{2} \)
47 \( 1 + 3.47iT - 47T^{2} \)
53 \( 1 - 3.95iT - 53T^{2} \)
59 \( 1 + (7.83 + 7.83i)T + 59iT^{2} \)
61 \( 1 + (-9.10 - 9.10i)T + 61iT^{2} \)
67 \( 1 - 9.29iT - 67T^{2} \)
71 \( 1 + 1.23iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + (0.761 - 0.761i)T - 79iT^{2} \)
83 \( 1 + 9.18iT - 83T^{2} \)
89 \( 1 + (-5.54 + 5.54i)T - 89iT^{2} \)
97 \( 1 + (-3.84 + 3.84i)T - 97iT^{2} \)
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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

−9.292317223511147682244965964689, −9.141577532369392116035230757219, −8.145966456273649385383358395189, −7.47516046001254784422385823074, −6.83021296815892767457325615678, −5.80832196964116154046611885855, −4.47232573143877419662288863311, −3.78451464010683898432489653009, −2.11379850241544261769406385883, −1.21325129917161813778707124362, 1.23512921588529045541685166414, 2.59331029405396461632208419126, 3.39482257210607002149496638637, 4.37350150034562308368182186341, 5.30067259194935536821515477097, 6.91690657684026262539445159039, 7.63422813086020290969154990314, 8.214621274960103664201847210438, 9.286836990989532035936720682346, 9.584286591975414737575167191929

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