Properties

Label 2-1110-185.184-c1-0-25
Degree $2$
Conductor $1110$
Sign $0.726 + 0.687i$
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 + (−1.32 + 1.79i)5-s i·6-s − 1.87i·7-s + 8-s − 9-s + (−1.32 + 1.79i)10-s − 1.51·11-s i·12-s + 4.78·13-s − 1.87i·14-s + (1.79 + 1.32i)15-s + 16-s + 3.99·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.5·4-s + (−0.594 + 0.804i)5-s − 0.408i·6-s − 0.707i·7-s + 0.353·8-s − 0.333·9-s + (−0.420 + 0.568i)10-s − 0.455·11-s − 0.288i·12-s + 1.32·13-s − 0.500i·14-s + (0.464 + 0.343i)15-s + 0.250·16-s + 0.968·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.726 + 0.687i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.363714535\)
\(L(\frac12)\) \(\approx\) \(2.363714535\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 1.79i)T \)
37 \( 1 + (-6.03 - 0.734i)T \)
good7 \( 1 + 1.87iT - 7T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 - 4.78T + 13T^{2} \)
17 \( 1 - 3.99T + 17T^{2} \)
19 \( 1 + 4.68iT - 19T^{2} \)
23 \( 1 - 6.51T + 23T^{2} \)
29 \( 1 + 0.200iT - 29T^{2} \)
31 \( 1 + 5.79iT - 31T^{2} \)
41 \( 1 - 4.10T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 5.15iT - 47T^{2} \)
53 \( 1 - 2.22iT - 53T^{2} \)
59 \( 1 - 13.2iT - 59T^{2} \)
61 \( 1 + 4.09iT - 61T^{2} \)
67 \( 1 + 5.76iT - 67T^{2} \)
71 \( 1 + 9.17T + 71T^{2} \)
73 \( 1 - 14.2iT - 73T^{2} \)
79 \( 1 - 5.95iT - 79T^{2} \)
83 \( 1 + 9.42iT - 83T^{2} \)
89 \( 1 - 5.47iT - 89T^{2} \)
97 \( 1 - 2.03T + 97T^{2} \)
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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

−10.00257418810323587520841660572, −8.698024186836573367687069309153, −7.78021936085600977295293113048, −7.18355655912338833027616701279, −6.47060402639498086803506990145, −5.56790194297602268556806541431, −4.37406789716160584973215595646, −3.44618111827238633069411799440, −2.65250326030319616812519497575, −0.981964054686185985015671581216, 1.36313181518439232958754855129, 3.06340584402841054449206323565, 3.75624021285170581551425461826, 4.82719922069023151649933422892, 5.47382146431071830265648832287, 6.24785115072271315353129234872, 7.58347588154652382290536530960, 8.366821327942192216518095616870, 8.991346103286866035493574648984, 10.00452424568328181996093846748

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