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Properties

Label 2-1176-8.5-c1-0-28
Degree $2$
Conductor $1176$
Sign $-0.841 - 0.539i$
Analytic cond. $9.39040$
Root an. cond. $3.06437$
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  + (1.06 + 0.925i)2-s + i·3-s + (0.285 + 1.97i)4-s − 1.80i·5-s + (−0.925 + 1.06i)6-s + (−1.52 + 2.38i)8-s − 9-s + (1.67 − 1.92i)10-s + 5.17i·11-s + (−1.97 + 0.285i)12-s − 0.840i·13-s + 1.80·15-s + (−3.83 + 1.13i)16-s + 4.91·17-s + (−1.06 − 0.925i)18-s + 5.63i·19-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)2-s + 0.577i·3-s + (0.142 + 0.989i)4-s − 0.806i·5-s + (−0.377 + 0.436i)6-s + (−0.539 + 0.841i)8-s − 0.333·9-s + (0.528 − 0.609i)10-s + 1.55i·11-s + (−0.571 + 0.0824i)12-s − 0.232i·13-s + 0.465·15-s + (−0.959 + 0.282i)16-s + 1.19·17-s + (−0.251 − 0.218i)18-s + 1.29i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.841 - 0.539i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.841 - 0.539i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.127721386\)
\(L(\frac12)\) \(\approx\) \(2.127721386\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 - 0.925i)T \)
3 \( 1 - iT \)
7 \( 1 \)
good5 \( 1 + 1.80iT - 5T^{2} \)
11 \( 1 - 5.17iT - 11T^{2} \)
13 \( 1 + 0.840iT - 13T^{2} \)
17 \( 1 - 4.91T + 17T^{2} \)
19 \( 1 - 5.63iT - 19T^{2} \)
23 \( 1 + 6.10T + 23T^{2} \)
29 \( 1 - 0.439iT - 29T^{2} \)
31 \( 1 + 7.33T + 31T^{2} \)
37 \( 1 - 5.26iT - 37T^{2} \)
41 \( 1 - 6.23T + 41T^{2} \)
43 \( 1 - 7.34iT - 43T^{2} \)
47 \( 1 + 5.67T + 47T^{2} \)
53 \( 1 - 1.33iT - 53T^{2} \)
59 \( 1 + 8.44iT - 59T^{2} \)
61 \( 1 + 5.51iT - 61T^{2} \)
67 \( 1 + 0.747iT - 67T^{2} \)
71 \( 1 - 9.08T + 71T^{2} \)
73 \( 1 - 7.41T + 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 - 1.45iT - 83T^{2} \)
89 \( 1 + 6.20T + 89T^{2} \)
97 \( 1 - 5.81T + 97T^{2} \)
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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

−9.838731080854834869267797779420, −9.406916830008677728154462488005, −8.082303772918839368387629379175, −7.86125925999193129448771546109, −6.65007492865622545525690943311, −5.65585252547801028784560545946, −5.00014651174486805592299789371, −4.21775432547349582700133761726, −3.37458378525392691311431741296, −1.87526855236097675980353333379, 0.70683080549140316482869911729, 2.20046882307075151443102563330, 3.13060967935357650885203630359, 3.86024609614059802405797793615, 5.32025218059077425800376237571, 5.95644189464681411960722043730, 6.75667063920026080152697798370, 7.60666142729218350788451144992, 8.721703944371532452565778063975, 9.569989963816172458433243439955

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