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Properties

Label 2-1110-111.80-c1-0-27
Degree $2$
Conductor $1110$
Sign $0.907 + 0.420i$
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 + 0.770i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (1.64 + 0.551i)6-s + 4.38·7-s + (0.707 − 0.707i)8-s + (1.81 − 2.39i)9-s − 1.00·10-s + 4.49·11-s + (−0.770 − 1.55i)12-s + (1.45 + 1.45i)13-s + (−3.09 − 3.09i)14-s + (−0.551 + 1.64i)15-s − 1.00·16-s + (1.09 − 1.09i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.895 + 0.445i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (0.670 + 0.225i)6-s + 1.65·7-s + (0.250 − 0.250i)8-s + (0.603 − 0.797i)9-s − 0.316·10-s + 1.35·11-s + (−0.222 − 0.447i)12-s + (0.403 + 0.403i)13-s + (−0.828 − 0.828i)14-s + (−0.142 + 0.423i)15-s − 0.250·16-s + (0.264 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.907 + 0.420i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.907 + 0.420i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.332514810\)
\(L(\frac12)\) \(\approx\) \(1.332514810\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.55 - 0.770i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (-0.167 + 6.08i)T \)
good7 \( 1 - 4.38T + 7T^{2} \)
11 \( 1 - 4.49T + 11T^{2} \)
13 \( 1 + (-1.45 - 1.45i)T + 13iT^{2} \)
17 \( 1 + (-1.09 + 1.09i)T - 17iT^{2} \)
19 \( 1 + (0.737 + 0.737i)T + 19iT^{2} \)
23 \( 1 + (1.56 - 1.56i)T - 23iT^{2} \)
29 \( 1 + (3.68 + 3.68i)T + 29iT^{2} \)
31 \( 1 + (0.303 - 0.303i)T - 31iT^{2} \)
41 \( 1 + 1.99T + 41T^{2} \)
43 \( 1 + (-0.568 - 0.568i)T + 43iT^{2} \)
47 \( 1 - 8.83iT - 47T^{2} \)
53 \( 1 - 8.07iT - 53T^{2} \)
59 \( 1 + (-9.81 + 9.81i)T - 59iT^{2} \)
61 \( 1 + (-0.496 + 0.496i)T - 61iT^{2} \)
67 \( 1 + 7.31iT - 67T^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 - 12.3iT - 73T^{2} \)
79 \( 1 + (2.04 + 2.04i)T + 79iT^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 + (-11.8 - 11.8i)T + 89iT^{2} \)
97 \( 1 + (0.766 + 0.766i)T + 97iT^{2} \)
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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

−9.674413696376628912955707842099, −9.221947107929244166761610306475, −8.328294965339168471093048011486, −7.38810877869994364280704408269, −6.35165309595039411518630459788, −5.41403961708947002425097243147, −4.47575727929305364263848280000, −3.82890898327286020473585611427, −1.90832779768080981009154727777, −1.06170519105911163483481721501, 1.16860330868811426490476951923, 1.92861289280382081672811050011, 3.98887675514566670831063962781, 5.04627027334995586594675823150, 5.75819324303484130019731441477, 6.60981125616221175662405523368, 7.33125224741906978218202924104, 8.210865503219733298526814280672, 8.836419840276544138376350571756, 10.10076902187425867760925844093

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