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Properties

Label 2-1110-37.36-c3-0-7
Degree $2$
Conductor $1110$
Sign $-0.790 - 0.612i$
Analytic cond. $65.4921$
Root an. cond. $8.09272$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 3·3-s − 4·4-s − 5i·5-s + 6i·6-s − 31.7·7-s − 8i·8-s + 9·9-s + 10·10-s + 2.18·11-s − 12·12-s − 61.0i·13-s − 63.5i·14-s − 15i·15-s + 16·16-s − 109. i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s − 1.71·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s + 0.0599·11-s − 0.288·12-s − 1.30i·13-s − 1.21i·14-s − 0.258i·15-s + 0.250·16-s − 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.790 - 0.612i$
Analytic conductor: \(65.4921\)
Root analytic conductor: \(8.09272\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :3/2),\ -0.790 - 0.612i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6920688772\)
\(L(\frac12)\) \(\approx\) \(0.6920688772\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 - 3T \)
5 \( 1 + 5iT \)
37 \( 1 + (-177. - 137. i)T \)
good7 \( 1 + 31.7T + 343T^{2} \)
11 \( 1 - 2.18T + 1.33e3T^{2} \)
13 \( 1 + 61.0iT - 2.19e3T^{2} \)
17 \( 1 + 109. iT - 4.91e3T^{2} \)
19 \( 1 + 114. iT - 6.85e3T^{2} \)
23 \( 1 - 175. iT - 1.21e4T^{2} \)
29 \( 1 - 284. iT - 2.43e4T^{2} \)
31 \( 1 - 273. iT - 2.97e4T^{2} \)
41 \( 1 + 159.T + 6.89e4T^{2} \)
43 \( 1 - 147. iT - 7.95e4T^{2} \)
47 \( 1 + 236.T + 1.03e5T^{2} \)
53 \( 1 + 245.T + 1.48e5T^{2} \)
59 \( 1 + 212. iT - 2.05e5T^{2} \)
61 \( 1 + 279. iT - 2.26e5T^{2} \)
67 \( 1 - 77.5T + 3.00e5T^{2} \)
71 \( 1 + 162.T + 3.57e5T^{2} \)
73 \( 1 - 235.T + 3.89e5T^{2} \)
79 \( 1 - 1.19e3iT - 4.93e5T^{2} \)
83 \( 1 + 885.T + 5.71e5T^{2} \)
89 \( 1 - 782. iT - 7.04e5T^{2} \)
97 \( 1 + 1.27e3iT - 9.12e5T^{2} \)
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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

−9.471247107041539517723080488010, −9.106846001238845092839587609107, −8.139257904839079825372968761837, −7.12234119195759466828839486349, −6.73601242807255535409473941577, −5.48310805121798897041244391470, −4.85032727301193709656058613401, −3.27632498614456820662357650907, −3.04200480720179646148230942264, −0.973728688032957253834992348269, 0.17941615839429438919941849190, 1.88766776032833831228399243220, 2.69845312835160037557750030013, 3.86709193301500496968890005704, 4.11811011830211669859219065246, 6.19289127788815227621200303579, 6.27660733513229533775936851798, 7.61340040347608516645498392520, 8.509574474747567760884239010776, 9.359836811104249701507401670698

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