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Properties

Label 2-1110-111.68-c1-0-3
Degree $2$
Conductor $1110$
Sign $0.723 - 0.689i$
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 + (−1.33 − 1.10i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (−1.72 + 0.159i)6-s − 3.17·7-s + (−0.707 − 0.707i)8-s + (0.549 + 2.94i)9-s − 1.00·10-s − 4.29·11-s + (−1.10 + 1.33i)12-s + (0.469 − 0.469i)13-s + (−2.24 + 2.24i)14-s + (0.159 + 1.72i)15-s − 1.00·16-s + (4.17 + 4.17i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.769 − 0.639i)3-s − 0.500i·4-s + (−0.316 − 0.316i)5-s + (−0.704 + 0.0650i)6-s − 1.19·7-s + (−0.250 − 0.250i)8-s + (0.183 + 0.983i)9-s − 0.316·10-s − 1.29·11-s + (−0.319 + 0.384i)12-s + (0.130 − 0.130i)13-s + (−0.599 + 0.599i)14-s + (0.0411 + 0.445i)15-s − 0.250·16-s + (1.01 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.723 - 0.689i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.723 - 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4937911995\)
\(L(\frac12)\) \(\approx\) \(0.4937911995\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 + 1.10i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-3.36 - 5.06i)T \)
good7 \( 1 + 3.17T + 7T^{2} \)
11 \( 1 + 4.29T + 11T^{2} \)
13 \( 1 + (-0.469 + 0.469i)T - 13iT^{2} \)
17 \( 1 + (-4.17 - 4.17i)T + 17iT^{2} \)
19 \( 1 + (-2.35 + 2.35i)T - 19iT^{2} \)
23 \( 1 + (-2.43 - 2.43i)T + 23iT^{2} \)
29 \( 1 + (3.28 - 3.28i)T - 29iT^{2} \)
31 \( 1 + (0.0756 + 0.0756i)T + 31iT^{2} \)
41 \( 1 + 6.28T + 41T^{2} \)
43 \( 1 + (-2.94 + 2.94i)T - 43iT^{2} \)
47 \( 1 - 0.554iT - 47T^{2} \)
53 \( 1 - 10.4iT - 53T^{2} \)
59 \( 1 + (5.42 + 5.42i)T + 59iT^{2} \)
61 \( 1 + (3.80 + 3.80i)T + 61iT^{2} \)
67 \( 1 + 7.22iT - 67T^{2} \)
71 \( 1 - 8.00iT - 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 + (2.36 - 2.36i)T - 79iT^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (2.62 - 2.62i)T - 89iT^{2} \)
97 \( 1 + (0.130 - 0.130i)T - 97iT^{2} \)
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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

−10.21818159077614263222045580485, −9.346180238368569313746453396906, −8.112401507557164713741674579899, −7.36652599208071874264888869494, −6.41289809528368070606790884757, −5.58114087312623921258855527236, −4.97852519432416914316109674024, −3.61873491867447244301548563144, −2.71988005537174230175189342916, −1.20776847904636828185521627600, 0.22253445153598913558190020893, 2.90654713014545656305048411325, 3.53976213355128308595534165670, 4.68804312700194531851839934747, 5.53535215455047698192294817987, 6.17602153188616181378228166136, 7.13879410059346513763975823583, 7.78850245002611846090821629567, 9.078257371830653347429050850385, 9.876665872936862902753675156700

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