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Properties

Label 2-1134-63.38-c1-0-14
Degree $2$
Conductor $1134$
Sign $0.690 - 0.723i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
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 − 4-s + (0.866 − 1.5i)5-s + (−2.5 + 0.866i)7-s − i·8-s + (1.5 + 0.866i)10-s + (−0.866 − 2.5i)14-s + 16-s + (−1.73 + 3i)17-s + (6 − 3.46i)19-s + (−0.866 + 1.5i)20-s + (5.19 + 3i)23-s + (1 + 1.73i)25-s + (2.5 − 0.866i)28-s + (7.79 + 4.5i)29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.387 − 0.670i)5-s + (−0.944 + 0.327i)7-s − 0.353i·8-s + (0.474 + 0.273i)10-s + (−0.231 − 0.668i)14-s + 0.250·16-s + (−0.420 + 0.727i)17-s + (1.37 − 0.794i)19-s + (−0.193 + 0.335i)20-s + (1.08 + 0.625i)23-s + (0.200 + 0.346i)25-s + (0.472 − 0.163i)28-s + (1.44 + 0.835i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.690 - 0.723i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.690 - 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.499397234\)
\(L(\frac12)\) \(\approx\) \(1.499397234\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.73 - 3i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6 + 3.46i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.79 - 4.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.73 - 3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + (2.59 + 1.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 - 14T + 67T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (10.5 + 6.06i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + (8.66 - 15i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6 + 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.567677213253050646101759963134, −9.109344389049275009888686299443, −8.430849810704030697259361666793, −7.23678785413703348163947322413, −6.66701968615863156889854001739, −5.58299091608529065642164586534, −5.10763409244156535404898720691, −3.85184067384947370926943337712, −2.74267906647261803193101578632, −1.00103493056925685697696194661, 0.912304857533416237477968961567, 2.61465838675511528612564514236, 3.14513429687092525675014953306, 4.31193199757985699995381258471, 5.38542151630930705020418797158, 6.46961552477754556325695534326, 7.05794904483952272791193211280, 8.180615065755679954507861310378, 9.209554422738179310525200721401, 9.872752260004682719637446782314

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