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Properties

Label 2-1176-8.5-c1-0-17
Degree $2$
Conductor $1176$
Sign $-0.239 - 0.970i$
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  + (−1.40 + 0.114i)2-s + i·3-s + (1.97 − 0.321i)4-s + 1.12i·5-s + (−0.114 − 1.40i)6-s + (−2.74 + 0.678i)8-s − 9-s + (−0.128 − 1.59i)10-s − 4.76i·11-s + (0.321 + 1.97i)12-s − 0.456i·13-s − 1.12·15-s + (3.79 − 1.26i)16-s − 0.415·17-s + (1.40 − 0.114i)18-s + 7.63i·19-s + ⋯
L(s)  = 1  + (−0.996 + 0.0806i)2-s + 0.577i·3-s + (0.986 − 0.160i)4-s + 0.504i·5-s + (−0.0465 − 0.575i)6-s + (−0.970 + 0.239i)8-s − 0.333·9-s + (−0.0407 − 0.503i)10-s − 1.43i·11-s + (0.0928 + 0.569i)12-s − 0.126i·13-s − 0.291·15-s + (0.948 − 0.317i)16-s − 0.100·17-s + (0.332 − 0.0268i)18-s + 1.75i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.239 - 0.970i$
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: \((2,\ 1176,\ (\ :1/2),\ -0.239 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8741855715\)
\(L(\frac12)\) \(\approx\) \(0.8741855715\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.114i)T \)
3 \( 1 - iT \)
7 \( 1 \)
good5 \( 1 - 1.12iT - 5T^{2} \)
11 \( 1 + 4.76iT - 11T^{2} \)
13 \( 1 + 0.456iT - 13T^{2} \)
17 \( 1 + 0.415T + 17T^{2} \)
19 \( 1 - 7.63iT - 19T^{2} \)
23 \( 1 - 1.58T + 23T^{2} \)
29 \( 1 - 6.72iT - 29T^{2} \)
31 \( 1 - 5.89T + 31T^{2} \)
37 \( 1 - 5.89iT - 37T^{2} \)
41 \( 1 - 0.415T + 41T^{2} \)
43 \( 1 - 9.43iT - 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 - 7.63iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 - 1.80iT - 61T^{2} \)
67 \( 1 + 8.09iT - 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 3.34T + 73T^{2} \)
79 \( 1 + 4.83T + 79T^{2} \)
83 \( 1 + 5.53iT - 83T^{2} \)
89 \( 1 + 4.92T + 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−10.05565855162799497859803334103, −9.206802460720388997137606796994, −8.353033709328003905509295285878, −7.899506181941754954784448894176, −6.60680564153412801382712997518, −6.09395721363960634261571503817, −5.05031897555447163018659292124, −3.47984868721251698315256884844, −2.89823713561261682044088141664, −1.25000245294838359676855441983, 0.57379120543433173168641922071, 1.90225289481434728338024218342, 2.76587161961034791897129857279, 4.36529908287522382714420346322, 5.36875698930992676912770402603, 6.72603389680132842150560518301, 7.00287896286839680289722213269, 7.990544080633801395000959688644, 8.742085356624166597868011714880, 9.460112957399456499775733422150

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