Properties

Label 2-1110-185.142-c1-0-6
Degree $2$
Conductor $1110$
Sign $0.537 - 0.843i$
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  i·2-s + (0.707 + 0.707i)3-s − 4-s + (−1.52 − 1.63i)5-s + (0.707 − 0.707i)6-s + (1.84 + 1.84i)7-s + i·8-s + 1.00i·9-s + (−1.63 + 1.52i)10-s + 3.01i·11-s + (−0.707 − 0.707i)12-s + 1.82i·13-s + (1.84 − 1.84i)14-s + (0.0778 − 2.23i)15-s + 16-s − 5.42·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−0.682 − 0.731i)5-s + (0.288 − 0.288i)6-s + (0.696 + 0.696i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.517 + 0.482i)10-s + 0.907i·11-s + (−0.204 − 0.204i)12-s + 0.504i·13-s + (0.492 − 0.492i)14-s + (0.0201 − 0.577i)15-s + 0.250·16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.537 - 0.843i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.537 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.217818856\)
\(L(\frac12)\) \(\approx\) \(1.217818856\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (1.52 + 1.63i)T \)
37 \( 1 + (-5.64 - 2.27i)T \)
good7 \( 1 + (-1.84 - 1.84i)T + 7iT^{2} \)
11 \( 1 - 3.01iT - 11T^{2} \)
13 \( 1 - 1.82iT - 13T^{2} \)
17 \( 1 + 5.42T + 17T^{2} \)
19 \( 1 + (0.509 + 0.509i)T + 19iT^{2} \)
23 \( 1 - 0.0216iT - 23T^{2} \)
29 \( 1 + (4.67 - 4.67i)T - 29iT^{2} \)
31 \( 1 + (-1.55 - 1.55i)T + 31iT^{2} \)
41 \( 1 - 6.35iT - 41T^{2} \)
43 \( 1 - 7.86iT - 43T^{2} \)
47 \( 1 + (0.176 + 0.176i)T + 47iT^{2} \)
53 \( 1 + (-2.05 + 2.05i)T - 53iT^{2} \)
59 \( 1 + (2.84 + 2.84i)T + 59iT^{2} \)
61 \( 1 + (-4.51 - 4.51i)T + 61iT^{2} \)
67 \( 1 + (-3.93 + 3.93i)T - 67iT^{2} \)
71 \( 1 - 9.00T + 71T^{2} \)
73 \( 1 + (-8.50 - 8.50i)T + 73iT^{2} \)
79 \( 1 + (2.16 + 2.16i)T + 79iT^{2} \)
83 \( 1 + (-0.925 + 0.925i)T - 83iT^{2} \)
89 \( 1 + (-1.03 + 1.03i)T - 89iT^{2} \)
97 \( 1 + 7.65T + 97T^{2} \)
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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

−9.767041026997415812979911885349, −9.206472551111625272716760970594, −8.508451804503099437219939136499, −7.86941748398891319453220633520, −6.71919672040343236119453224879, −5.20825639772164242483966219317, −4.63157579410233728154708664503, −3.92271507497406885343818519345, −2.56728740918844535641581064475, −1.58532469321838416267068285363, 0.51833207419621138922464947685, 2.36331795352749685404484285339, 3.66104389597101867611225379918, 4.32698071563844230139007940422, 5.64150089390469184342192064899, 6.57197980125270367239842437998, 7.30365383856335589207210925235, 7.983046958821146004574088765070, 8.510024459617762346295565132496, 9.527736159196307337485925873383

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