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Properties

Label 2-156-13.10-c1-0-2
Degree $2$
Conductor $156$
Sign $-0.107 + 0.994i$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
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  + (−0.5 − 0.866i)3-s − 2.41i·5-s + (−3.58 − 2.07i)7-s + (−0.499 + 0.866i)9-s + (3 − 1.73i)11-s + (1.5 + 3.27i)13-s + (−2.08 + 1.20i)15-s + (3.08 − 5.35i)17-s + (−3 − 1.73i)19-s + 4.14i·21-s + (1 + 1.73i)23-s − 0.821·25-s + 0.999·27-s + (4.08 + 7.08i)29-s + 7.60i·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s − 1.07i·5-s + (−1.35 − 0.783i)7-s + (−0.166 + 0.288i)9-s + (0.904 − 0.522i)11-s + (0.416 + 0.909i)13-s + (−0.539 + 0.311i)15-s + (0.749 − 1.29i)17-s + (−0.688 − 0.397i)19-s + 0.904i·21-s + (0.208 + 0.361i)23-s − 0.164·25-s + 0.192·27-s + (0.759 + 1.31i)29-s + 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.107 + 0.994i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ -0.107 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.604169 - 0.672776i\)
\(L(\frac12)\) \(\approx\) \(0.604169 - 0.672776i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-1.5 - 3.27i)T \)
good5 \( 1 + 2.41iT - 5T^{2} \)
7 \( 1 + (3.58 + 2.07i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.08 + 5.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.08 - 7.08i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.60iT - 31T^{2} \)
37 \( 1 + (-0.910 + 0.525i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.08 + 2.93i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.410 + 0.711i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + (-1.17 - 0.680i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.58 + 2.07i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 1.73i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 - 11.7iT - 83T^{2} \)
89 \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.76 + 1.02i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.63542491920370179335042059816, −11.94220474938429936559650112401, −10.72666758155861706231801466833, −9.398049938313882028041960117484, −8.757609311142942260007177683704, −7.12438838360955203015998278324, −6.40095072983450598738511208066, −4.91801569115448637966531894565, −3.46795003925997356317729915601, −0.984190949680894010672233780496, 2.84363626403562000721962077990, 3.98459873337063147491050668106, 6.07679891007316324166050458079, 6.35042230918559717612293869896, 8.032716176996913019862520911085, 9.447052467538794767492652999976, 10.13855176634361831893891498911, 11.05695514709130844748536269475, 12.28370766636325541977876847689, 12.96078973274210685023345368946

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