Properties

Label 2-1134-63.38-c1-0-26
Degree $2$
Conductor $1134$
Sign $0.906 + 0.421i$
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 + (2.09 − 3.62i)5-s + (2.62 − 0.358i)7-s i·8-s + (3.62 + 2.09i)10-s + (2.59 − 1.5i)11-s + (−2.12 + 1.22i)13-s + (0.358 + 2.62i)14-s + 16-s + (−0.507 + 0.878i)17-s + (−0.878 + 0.507i)19-s + (−2.09 + 3.62i)20-s + (1.5 + 2.59i)22-s + (3.67 + 2.12i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.935 − 1.61i)5-s + (0.990 − 0.135i)7-s − 0.353i·8-s + (1.14 + 0.661i)10-s + (0.783 − 0.452i)11-s + (−0.588 + 0.339i)13-s + (0.0958 + 0.700i)14-s + 0.250·16-s + (−0.123 + 0.213i)17-s + (−0.201 + 0.116i)19-s + (−0.467 + 0.809i)20-s + (0.319 + 0.553i)22-s + (0.766 + 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.906 + 0.421i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.906 + 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.023517576\)
\(L(\frac12)\) \(\approx\) \(2.023517576\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 + 0.358i)T \)
good5 \( 1 + (-2.09 + 3.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.12 - 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.507 - 0.878i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.878 - 0.507i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.67 - 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.07 - 0.621i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.61iT - 31T^{2} \)
37 \( 1 + (4.12 + 7.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.01 + 1.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.12 - 7.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.01T + 47T^{2} \)
53 \( 1 + (-1.07 - 0.621i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 - 5.91iT - 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (-7.24 - 4.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + (-1.58 + 2.74i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.25 - 1.88i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.381993264673161030249356595844, −8.914530585412244381533456969047, −8.271889445561445792398737531090, −7.34891324781678848848010287674, −6.23022355854124706435367024780, −5.44313591373444305418836203375, −4.79645508841850714777518476376, −4.01293905825042585200801203311, −2.00197412349096806972023762743, −0.980411682521153323900069134756, 1.61677208438451044090099080924, 2.48116113090829759941842861093, 3.36212926535160922796774501589, 4.66633461835822587684707927957, 5.51972678500407866692770604530, 6.69060614815535879890166818569, 7.17958797887611984017895073560, 8.399895300320080320385674861909, 9.266405376959575739430538673763, 10.21053812304436736248063465004

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