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Properties

Label 2-234-13.2-c2-0-2
Degree $2$
Conductor $234$
Sign $-0.793 - 0.608i$
Analytic cond. $6.37603$
Root an. cond. $2.52508$
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  + (1.36 + 0.366i)2-s + (1.73 + i)4-s + (−3.77 + 3.77i)5-s + (−9.91 + 2.65i)7-s + (1.99 + 2i)8-s + (−6.53 + 3.77i)10-s + (−2.71 + 10.1i)11-s + (−8.18 − 10.0i)13-s − 14.5·14-s + (1.99 + 3.46i)16-s + (−4.23 − 2.44i)17-s + (6.83 + 25.5i)19-s + (−10.3 + 2.76i)20-s + (−7.40 + 12.8i)22-s + (17.2 − 9.97i)23-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.433 + 0.250i)4-s + (−0.754 + 0.754i)5-s + (−1.41 + 0.379i)7-s + (0.249 + 0.250i)8-s + (−0.653 + 0.377i)10-s + (−0.246 + 0.919i)11-s + (−0.629 − 0.776i)13-s − 1.03·14-s + (0.124 + 0.216i)16-s + (−0.249 − 0.143i)17-s + (0.359 + 1.34i)19-s + (−0.515 + 0.138i)20-s + (−0.336 + 0.583i)22-s + (0.751 − 0.433i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.793 - 0.608i$
Analytic conductor: \(6.37603\)
Root analytic conductor: \(2.52508\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1),\ -0.793 - 0.608i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.372609 + 1.09908i\)
\(L(\frac12)\) \(\approx\) \(0.372609 + 1.09908i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 \)
13 \( 1 + (8.18 + 10.0i)T \)
good5 \( 1 + (3.77 - 3.77i)T - 25iT^{2} \)
7 \( 1 + (9.91 - 2.65i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (2.71 - 10.1i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (4.23 + 2.44i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.83 - 25.5i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-17.2 + 9.97i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-7.15 - 12.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-19.0 + 19.0i)T - 961iT^{2} \)
37 \( 1 + (15.7 - 58.6i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-4.83 - 1.29i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (10.3 + 5.98i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-7.59 - 7.59i)T + 2.20e3iT^{2} \)
53 \( 1 + 77.0T + 2.80e3T^{2} \)
59 \( 1 + (-60.7 + 16.2i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-28.1 + 48.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-5.90 - 1.58i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-14.7 - 55.2i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-12.7 - 12.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 7.98T + 6.24e3T^{2} \)
83 \( 1 + (-35.8 + 35.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (20.9 - 78.2i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (14.1 + 52.9i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−12.48942827004608062896038793990, −11.63148448372788101022945827361, −10.38083722982226693433648961786, −9.681325538428329901224066543960, −8.084365977506067500647349740104, −7.12408627989045902114946968097, −6.34329351279787405467267054623, −5.02627076530616436354339244585, −3.57257279253405429913361839705, −2.73370430079253983732814759687, 0.47821109105835118344601274821, 2.88746123738766675247410654313, 3.99820066115675214998323749430, 5.08823884170934058549249931603, 6.44937077987620085727846663854, 7.32496340358601313847908254531, 8.742626318198404571754373447055, 9.613459280478577236642870255247, 10.84379529521781832537821922880, 11.74122800211251234476851562457

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