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Properties

Label 2-1134-63.5-c1-0-12
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: $\chi_{1134} (215, \cdot )$
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

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

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