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Properties

Label 2-1110-185.184-c1-0-14
Degree $2$
Conductor $1110$
Sign $0.999 + 0.00296i$
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 − i·3-s + 4-s + (2.20 + 0.347i)5-s + i·6-s + 2.08i·7-s − 8-s − 9-s + (−2.20 − 0.347i)10-s + 6.52·11-s − i·12-s + 1.53·13-s − 2.08i·14-s + (0.347 − 2.20i)15-s + 16-s − 5.28·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577i·3-s + 0.5·4-s + (0.987 + 0.155i)5-s + 0.408i·6-s + 0.786i·7-s − 0.353·8-s − 0.333·9-s + (−0.698 − 0.109i)10-s + 1.96·11-s − 0.288i·12-s + 0.425·13-s − 0.556i·14-s + (0.0897 − 0.570i)15-s + 0.250·16-s − 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.999 + 0.00296i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.999 + 0.00296i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.547412076\)
\(L(\frac12)\) \(\approx\) \(1.547412076\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (-2.20 - 0.347i)T \)
37 \( 1 + (0.927 + 6.01i)T \)
good7 \( 1 - 2.08iT - 7T^{2} \)
11 \( 1 - 6.52T + 11T^{2} \)
13 \( 1 - 1.53T + 13T^{2} \)
17 \( 1 + 5.28T + 17T^{2} \)
19 \( 1 + 4.07iT - 19T^{2} \)
23 \( 1 - 1.43T + 23T^{2} \)
29 \( 1 - 6.89iT - 29T^{2} \)
31 \( 1 - 8.19iT - 31T^{2} \)
41 \( 1 - 6.76T + 41T^{2} \)
43 \( 1 + 4.46T + 43T^{2} \)
47 \( 1 - 9.83iT - 47T^{2} \)
53 \( 1 - 6.51iT - 53T^{2} \)
59 \( 1 + 11.0iT - 59T^{2} \)
61 \( 1 + 4.91iT - 61T^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 + 6.07T + 71T^{2} \)
73 \( 1 + 13.6iT - 73T^{2} \)
79 \( 1 - 2.76iT - 79T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 - 7.15iT - 89T^{2} \)
97 \( 1 - 19.0T + 97T^{2} \)
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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

−9.333775988469449495547316655467, −9.146574546501185579242376377760, −8.574846593915062844840742922983, −7.11477067442713949633540582709, −6.58080030775823098102485343848, −6.01032618974146831704121710631, −4.80828904458469093812943395755, −3.24809670033300199162584186280, −2.12343142299748424460401818551, −1.26100462576275456629050410779, 1.05904580960300380196444847083, 2.20173898567039834712022475372, 3.75996224446591987958254395147, 4.42714946147447425337754695899, 5.92064451675730303519502086451, 6.41335673990460887069966987488, 7.32120645425558156187026441672, 8.639284937119836962137510451106, 8.996086573596698291090634404334, 9.966024511151202217256427081761

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