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Properties

Label 2-1176-56.27-c1-0-36
Degree $2$
Conductor $1176$
Sign $0.970 - 0.240i$
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  + (−1.32 + 0.507i)2-s + i·3-s + (1.48 − 1.33i)4-s − 0.451·5-s + (−0.507 − 1.32i)6-s + (−1.28 + 2.52i)8-s − 9-s + (0.595 − 0.228i)10-s + 0.720·11-s + (1.33 + 1.48i)12-s + 3.48·13-s − 0.451i·15-s + (0.415 − 3.97i)16-s − 4.10i·17-s + (1.32 − 0.507i)18-s − 4.59i·19-s + ⋯
L(s)  = 1  + (−0.933 + 0.358i)2-s + 0.577i·3-s + (0.742 − 0.669i)4-s − 0.201·5-s + (−0.206 − 0.538i)6-s + (−0.453 + 0.891i)8-s − 0.333·9-s + (0.188 − 0.0723i)10-s + 0.217·11-s + (0.386 + 0.428i)12-s + 0.967·13-s − 0.116i·15-s + (0.103 − 0.994i)16-s − 0.996i·17-s + (0.311 − 0.119i)18-s − 1.05i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.970 - 0.240i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ 0.970 - 0.240i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.008554012\)
\(L(\frac12)\) \(\approx\) \(1.008554012\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 0.507i)T \)
3 \( 1 - iT \)
7 \( 1 \)
good5 \( 1 + 0.451T + 5T^{2} \)
11 \( 1 - 0.720T + 11T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 + 4.10iT - 17T^{2} \)
19 \( 1 + 4.59iT - 19T^{2} \)
23 \( 1 + 0.0530iT - 23T^{2} \)
29 \( 1 - 7.85iT - 29T^{2} \)
31 \( 1 - 9.16T + 31T^{2} \)
37 \( 1 + 8.67iT - 37T^{2} \)
41 \( 1 + 3.94iT - 41T^{2} \)
43 \( 1 - 5.17T + 43T^{2} \)
47 \( 1 + 0.920T + 47T^{2} \)
53 \( 1 + 3.13iT - 53T^{2} \)
59 \( 1 + 5.61iT - 59T^{2} \)
61 \( 1 - 5.09T + 61T^{2} \)
67 \( 1 - 9.86T + 67T^{2} \)
71 \( 1 - 11.1iT - 71T^{2} \)
73 \( 1 + 3.84iT - 73T^{2} \)
79 \( 1 - 9.71iT - 79T^{2} \)
83 \( 1 + 9.53iT - 83T^{2} \)
89 \( 1 - 14.5iT - 89T^{2} \)
97 \( 1 + 5.14iT - 97T^{2} \)
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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

−9.650341427731362897510410338791, −8.989913274011288378819967601725, −8.381804982346141071861604525505, −7.38319962304240785340973028133, −6.63694146256597120441187056218, −5.68367340555059509614795867627, −4.79486237108846615158975681671, −3.55303810170178626391490228163, −2.37342723682629685400391380831, −0.75976142794141418519822557627, 1.03738561245899252269966795734, 2.07768018292361344815939822628, 3.33948932212965040546683729960, 4.23605549898814117744277739226, 6.11496850454093921716278591892, 6.29769253136818576957794230405, 7.63567162232977100639237398217, 8.132506883856789208925345811002, 8.739679082659854198221485470842, 9.871313660697011060910284182545

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