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Properties

Label 2-156-3.2-c2-0-0
Degree $2$
Conductor $156$
Sign $-0.446 - 0.894i$
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  + (−1.33 − 2.68i)3-s + 5.74i·5-s − 9.51·7-s + (−5.41 + 7.19i)9-s + 4.60i·11-s + 3.60·13-s + (15.4 − 7.69i)15-s + 29.3i·17-s − 25.4·19-s + (12.7 + 25.5i)21-s + 2.28i·23-s − 7.98·25-s + (26.5 + 4.89i)27-s − 22.5i·29-s − 24.0·31-s + ⋯
L(s)  = 1  + (−0.446 − 0.894i)3-s + 1.14i·5-s − 1.35·7-s + (−0.601 + 0.798i)9-s + 0.418i·11-s + 0.277·13-s + (1.02 − 0.512i)15-s + 1.72i·17-s − 1.34·19-s + (0.606 + 1.21i)21-s + 0.0992i·23-s − 0.319·25-s + (0.983 + 0.181i)27-s − 0.776i·29-s − 0.775·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.446 - 0.894i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ -0.446 - 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.270801 + 0.437743i\)
\(L(\frac12)\) \(\approx\) \(0.270801 + 0.437743i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.33 + 2.68i)T \)
13 \( 1 - 3.60T \)
good5 \( 1 - 5.74iT - 25T^{2} \)
7 \( 1 + 9.51T + 49T^{2} \)
11 \( 1 - 4.60iT - 121T^{2} \)
17 \( 1 - 29.3iT - 289T^{2} \)
19 \( 1 + 25.4T + 361T^{2} \)
23 \( 1 - 2.28iT - 529T^{2} \)
29 \( 1 + 22.5iT - 841T^{2} \)
31 \( 1 + 24.0T + 961T^{2} \)
37 \( 1 + 60.0T + 1.36e3T^{2} \)
41 \( 1 + 59.7iT - 1.68e3T^{2} \)
43 \( 1 - 67.5T + 1.84e3T^{2} \)
47 \( 1 - 5.77iT - 2.20e3T^{2} \)
53 \( 1 - 77.6iT - 2.80e3T^{2} \)
59 \( 1 + 6.84iT - 3.48e3T^{2} \)
61 \( 1 - 0.585T + 3.72e3T^{2} \)
67 \( 1 + 12.2T + 4.48e3T^{2} \)
71 \( 1 + 57.5iT - 5.04e3T^{2} \)
73 \( 1 + 26.0T + 5.32e3T^{2} \)
79 \( 1 + 5.70T + 6.24e3T^{2} \)
83 \( 1 - 127. iT - 6.88e3T^{2} \)
89 \( 1 + 3.63iT - 7.92e3T^{2} \)
97 \( 1 - 169.T + 9.40e3T^{2} \)
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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

−12.82826434938406155117373020697, −12.31775867185788504715667987817, −10.80071005892213118674930480984, −10.41536320009066181692189108408, −8.849005441000714797040057289264, −7.46888072079193186793766082789, −6.53741212035221771698081796880, −5.96705693926691516447551185378, −3.73175291439739932914992623450, −2.24501103200107494890343728700, 0.32557321869070587825594883821, 3.24659518277095055269310400451, 4.57801102791016603760404589162, 5.64790161592643771375399630078, 6.80364141199976245735860102615, 8.703434667727961345194075111711, 9.260441627977356728511260460959, 10.24097573977787656566820002742, 11.36292615912629780000636950905, 12.45351295343984479364547196769

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