Properties

Label 2-234-13.6-c2-0-6
Degree $2$
Conductor $234$
Sign $0.804 - 0.594i$
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  + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + (5.04 − 5.04i)5-s + (−1.34 + 5.02i)7-s + (−2 − 1.99i)8-s + (8.74 + 5.04i)10-s + (7.32 − 1.96i)11-s + (12.9 + 1.42i)13-s − 7.36·14-s + (1.99 − 3.46i)16-s + (13.9 − 8.08i)17-s + (9.87 + 2.64i)19-s + (−3.69 + 13.7i)20-s + (5.36 + 9.28i)22-s + (8.29 + 4.78i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.250i)4-s + (1.00 − 1.00i)5-s + (−0.192 + 0.718i)7-s + (−0.250 − 0.249i)8-s + (0.874 + 0.504i)10-s + (0.665 − 0.178i)11-s + (0.993 + 0.109i)13-s − 0.525·14-s + (0.124 − 0.216i)16-s + (0.823 − 0.475i)17-s + (0.519 + 0.139i)19-s + (−0.184 + 0.689i)20-s + (0.243 + 0.422i)22-s + (0.360 + 0.208i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.804 - 0.594i$
Analytic conductor: \(6.37603\)
Root analytic conductor: \(2.52508\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1),\ 0.804 - 0.594i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.91309 + 0.630477i\)
\(L(\frac12)\) \(\approx\) \(1.91309 + 0.630477i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 \)
13 \( 1 + (-12.9 - 1.42i)T \)
good5 \( 1 + (-5.04 + 5.04i)T - 25iT^{2} \)
7 \( 1 + (1.34 - 5.02i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-7.32 + 1.96i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (-13.9 + 8.08i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-9.87 - 2.64i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (-8.29 - 4.78i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (16.5 - 28.7i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (34.2 - 34.2i)T - 961iT^{2} \)
37 \( 1 + (-63.2 + 16.9i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (14.0 + 52.6i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (40.7 - 23.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (47.8 + 47.8i)T + 2.20e3iT^{2} \)
53 \( 1 + 67.5T + 2.80e3T^{2} \)
59 \( 1 + (-19.4 + 72.7i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (35.2 + 61.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-11.1 - 41.7i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-106. - 28.6i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (36.8 + 36.8i)T + 5.32e3iT^{2} \)
79 \( 1 + 13.3T + 6.24e3T^{2} \)
83 \( 1 + (93.1 - 93.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (48.6 - 13.0i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (52.5 + 14.0i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−12.38348837061962095462028042408, −11.20088633251522130127208347100, −9.615240370762823262709340527705, −9.137978509544044955403462673813, −8.265202615897118741101968781991, −6.80121115548165874541389957415, −5.71253803402394788286519035165, −5.15024145694666370882662666770, −3.47971197579200124111141662348, −1.43027874335181753694602157070, 1.43104049532915552620874427416, 2.98002214966656248199241847507, 4.07359467749327303983023183852, 5.78104556124072354690322328244, 6.55694607013509397549126006833, 7.86137437130883245837224307612, 9.420134679035041116393280831481, 9.982995489660396825792324305760, 10.90632843583753425202963737042, 11.59591413976478076058630843667

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