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Properties

Label 2-156-4.3-c2-0-23
Degree $2$
Conductor $156$
Sign $-0.999 + 0.000136i$
Analytic cond. $4.25069$
Root an. cond. $2.06172$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 1.41i)2-s − 1.73i·3-s + (−0.000546 − 3.99i)4-s − 9.54·5-s + (−2.44 − 2.44i)6-s + 4.66i·7-s + (−5.65 − 5.65i)8-s − 2.99·9-s + (−13.4 + 13.4i)10-s − 13.9i·11-s + (−6.92 + 0.000946i)12-s + 3.60·13-s + (6.60 + 6.60i)14-s + 16.5i·15-s + (−15.9 + 0.00437i)16-s + 13.3·17-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 0.577i·3-s + (−0.000136 − 0.999i)4-s − 1.90·5-s + (−0.408 − 0.408i)6-s + 0.666i·7-s + (−0.707 − 0.706i)8-s − 0.333·9-s + (−1.34 + 1.34i)10-s − 1.26i·11-s + (−0.577 + 7.88e−5i)12-s + 0.277·13-s + (0.471 + 0.471i)14-s + 1.10i·15-s + (−0.999 + 0.000273i)16-s + 0.787·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.000136i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.000136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.999 + 0.000136i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ -0.999 + 0.000136i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(7.53561\times10^{-5} - 1.10312i\)
\(L(\frac12)\) \(\approx\) \(7.53561\times10^{-5} - 1.10312i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 1.41i)T \)
3 \( 1 + 1.73iT \)
13 \( 1 - 3.60T \)
good5 \( 1 + 9.54T + 25T^{2} \)
7 \( 1 - 4.66iT - 49T^{2} \)
11 \( 1 + 13.9iT - 121T^{2} \)
17 \( 1 - 13.3T + 289T^{2} \)
19 \( 1 + 11.4iT - 361T^{2} \)
23 \( 1 + 33.8iT - 529T^{2} \)
29 \( 1 + 32.7T + 841T^{2} \)
31 \( 1 + 5.21iT - 961T^{2} \)
37 \( 1 + 36.7T + 1.36e3T^{2} \)
41 \( 1 - 13.6T + 1.68e3T^{2} \)
43 \( 1 - 3.89iT - 1.84e3T^{2} \)
47 \( 1 - 54.8iT - 2.20e3T^{2} \)
53 \( 1 - 68.4T + 2.80e3T^{2} \)
59 \( 1 + 86.6iT - 3.48e3T^{2} \)
61 \( 1 - 67.1T + 3.72e3T^{2} \)
67 \( 1 + 27.0iT - 4.48e3T^{2} \)
71 \( 1 + 56.1iT - 5.04e3T^{2} \)
73 \( 1 + 19.3T + 5.32e3T^{2} \)
79 \( 1 - 85.7iT - 6.24e3T^{2} \)
83 \( 1 - 132. iT - 6.88e3T^{2} \)
89 \( 1 + 72.4T + 7.92e3T^{2} \)
97 \( 1 - 129.T + 9.40e3T^{2} \)
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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

−12.19244063735792088745636754920, −11.40750734914256072399927071257, −10.81124141457011133415239688426, −8.932140562987991138266698376840, −8.081891667102457465092946941975, −6.74338698019397579862072441525, −5.42089283295712777438080979854, −3.95501595267575520884958056466, −2.91024190156924723811481008052, −0.56630543998660493357640203969, 3.58184689804548811494802805943, 4.09302729868438726208304503137, 5.31493843169922098826586382932, 7.22355025491643663961108960524, 7.58132855578902590420861346360, 8.778890990588612185314541387464, 10.30627867401900564376351233216, 11.58399836470313450824880286114, 12.10271688784080233187345226258, 13.21740091235933193531040244462

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