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Properties

Label 2-1110-111.68-c1-0-12
Degree $2$
Conductor $1110$
Sign $-0.734 - 0.678i$
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 + (−0.396 − 1.68i)6-s − 2.79·7-s + (0.707 + 0.707i)8-s + (−1.33 − 2.68i)9-s − 1.00·10-s + 4.26·11-s + (1.47 + 0.912i)12-s + (1.77 − 1.77i)13-s + (1.97 − 1.97i)14-s + (−1.68 + 0.396i)15-s − 1.00·16-s + (5.48 + 5.48i)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 + (−0.161 − 0.688i)6-s − 1.05·7-s + (0.250 + 0.250i)8-s + (−0.445 − 0.895i)9-s − 0.316·10-s + 1.28·11-s + (0.425 + 0.263i)12-s + (0.492 − 0.492i)13-s + (0.528 − 0.528i)14-s + (−0.435 + 0.102i)15-s − 0.250·16-s + (1.33 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.734 - 0.678i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9247558138\)
\(L(\frac12)\) \(\approx\) \(0.9247558138\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.97 - 5.30i)T \)
good7 \( 1 + 2.79T + 7T^{2} \)
11 \( 1 - 4.26T + 11T^{2} \)
13 \( 1 + (-1.77 + 1.77i)T - 13iT^{2} \)
17 \( 1 + (-5.48 - 5.48i)T + 17iT^{2} \)
19 \( 1 + (-0.485 + 0.485i)T - 19iT^{2} \)
23 \( 1 + (-1.36 - 1.36i)T + 23iT^{2} \)
29 \( 1 + (7.02 - 7.02i)T - 29iT^{2} \)
31 \( 1 + (3.79 + 3.79i)T + 31iT^{2} \)
41 \( 1 - 7.88T + 41T^{2} \)
43 \( 1 + (3.72 - 3.72i)T - 43iT^{2} \)
47 \( 1 + 2.41iT - 47T^{2} \)
53 \( 1 - 2.32iT - 53T^{2} \)
59 \( 1 + (4.84 + 4.84i)T + 59iT^{2} \)
61 \( 1 + (-3.08 - 3.08i)T + 61iT^{2} \)
67 \( 1 + 0.631iT - 67T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 - 8.24iT - 73T^{2} \)
79 \( 1 + (4.17 - 4.17i)T - 79iT^{2} \)
83 \( 1 - 12.6iT - 83T^{2} \)
89 \( 1 + (1.13 - 1.13i)T - 89iT^{2} \)
97 \( 1 + (-3.82 + 3.82i)T - 97iT^{2} \)
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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

−9.941783443039308481842425808198, −9.468449836141004642000453913730, −8.793358558785860870847700623629, −7.63579554490787924681293248016, −6.53650979551321435901695440371, −6.07385310559841402151138483607, −5.34833675322472207785048057251, −3.87880419591629040923486820148, −3.29462551055338008947298107602, −1.24998534171413282447470485933, 0.60418440388070672202007847282, 1.69857365180586870531777654281, 2.97921451820358750031084051309, 4.07207084497845777297393402782, 5.48792269165637597122823329404, 6.25876551761308413132315921244, 7.06507234402808226494066061238, 7.77631638159518893091066643833, 9.056332854761954311738336437960, 9.378906773646668350100669029247

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