Properties

Label 2-1110-185.68-c1-0-6
Degree $2$
Conductor $1110$
Sign $-0.787 - 0.615i$
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  − 2-s + (−0.707 + 0.707i)3-s + 4-s + (−0.866 + 2.06i)5-s + (0.707 − 0.707i)6-s + (−0.662 + 0.662i)7-s − 8-s − 1.00i·9-s + (0.866 − 2.06i)10-s + 3.51i·11-s + (−0.707 + 0.707i)12-s + 6.87·13-s + (0.662 − 0.662i)14-s + (−0.844 − 2.07i)15-s + 16-s + 2.17i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (−0.387 + 0.921i)5-s + (0.288 − 0.288i)6-s + (−0.250 + 0.250i)7-s − 0.353·8-s − 0.333i·9-s + (0.273 − 0.651i)10-s + 1.06i·11-s + (−0.204 + 0.204i)12-s + 1.90·13-s + (0.176 − 0.176i)14-s + (−0.218 − 0.534i)15-s + 0.250·16-s + 0.527i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.787 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.787 - 0.615i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.787 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7966429543\)
\(L(\frac12)\) \(\approx\) \(0.7966429543\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.866 - 2.06i)T \)
37 \( 1 + (5.95 - 1.24i)T \)
good7 \( 1 + (0.662 - 0.662i)T - 7iT^{2} \)
11 \( 1 - 3.51iT - 11T^{2} \)
13 \( 1 - 6.87T + 13T^{2} \)
17 \( 1 - 2.17iT - 17T^{2} \)
19 \( 1 + (1.01 + 1.01i)T + 19iT^{2} \)
23 \( 1 - 9.57T + 23T^{2} \)
29 \( 1 + (6.49 - 6.49i)T - 29iT^{2} \)
31 \( 1 + (-2.38 - 2.38i)T + 31iT^{2} \)
41 \( 1 - 3.47iT - 41T^{2} \)
43 \( 1 - 2.74T + 43T^{2} \)
47 \( 1 + (-3.59 + 3.59i)T - 47iT^{2} \)
53 \( 1 + (2.93 + 2.93i)T + 53iT^{2} \)
59 \( 1 + (3.45 + 3.45i)T + 59iT^{2} \)
61 \( 1 + (9.99 + 9.99i)T + 61iT^{2} \)
67 \( 1 + (1.14 + 1.14i)T + 67iT^{2} \)
71 \( 1 + 7.32T + 71T^{2} \)
73 \( 1 + (0.292 - 0.292i)T - 73iT^{2} \)
79 \( 1 + (-7.95 - 7.95i)T + 79iT^{2} \)
83 \( 1 + (1.46 + 1.46i)T + 83iT^{2} \)
89 \( 1 + (6.43 - 6.43i)T - 89iT^{2} \)
97 \( 1 + 0.301iT - 97T^{2} \)
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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

−10.34253896194134439669895884927, −9.275928076680671557750010479662, −8.739880829219058099761221217765, −7.65264680290486700310149646162, −6.78014817075995029508937948417, −6.27455760548615363351672168655, −5.09609641175702767481643015589, −3.81458627450702843029826680902, −3.02444537626859866075993130877, −1.49135154614572352778357341868, 0.53200132611209224297353088596, 1.41126939299795625911107540055, 3.15733717831205014314321182481, 4.17639537161670799610443590109, 5.54105243347720167507378065258, 6.11218647693178956521246960714, 7.16906499669907977292116622905, 7.972603230542320031698178807603, 8.829960540852821145947816256353, 9.134728569106580504582438256716

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