↑

Properties

Label 2-234-117.50-c1-0-3
Degree $2$
Conductor $234$
Sign $0.620 - 0.784i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)2-s + (0.675 + 1.59i)3-s + (−0.866 + 0.499i)4-s + (0.435 + 1.62i)5-s + (1.36 − 1.06i)6-s + (−0.212 + 0.212i)7-s + (0.707 + 0.707i)8-s + (−2.08 + 2.15i)9-s + (1.45 − 0.842i)10-s + (−1.53 + 0.411i)11-s + (−1.38 − 1.04i)12-s + (2.24 + 2.82i)13-s + (0.260 + 0.150i)14-s + (−2.29 + 1.79i)15-s + (0.500 − 0.866i)16-s + (−1.26 + 2.19i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.390 + 0.920i)3-s + (−0.433 + 0.249i)4-s + (0.194 + 0.727i)5-s + (0.557 − 0.434i)6-s + (−0.0803 + 0.0803i)7-s + (0.249 + 0.249i)8-s + (−0.695 + 0.718i)9-s + (0.461 − 0.266i)10-s + (−0.462 + 0.124i)11-s + (−0.399 − 0.301i)12-s + (0.622 + 0.782i)13-s + (0.0695 + 0.0401i)14-s + (−0.593 + 0.463i)15-s + (0.125 − 0.216i)16-s + (−0.306 + 0.531i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.620 - 0.784i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.620 - 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06800 + 0.517165i\)
\(L(\frac12)\) \(\approx\) \(1.06800 + 0.517165i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (-0.675 - 1.59i)T \)
13 \( 1 + (-2.24 - 2.82i)T \)
good5 \( 1 + (-0.435 - 1.62i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.212 - 0.212i)T - 7iT^{2} \)
11 \( 1 + (1.53 - 0.411i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.26 - 2.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.16 + 1.11i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 1.32T + 23T^{2} \)
29 \( 1 + (6.48 + 3.74i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-10.3 + 2.78i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.37 + 0.637i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.66 + 2.66i)T - 41iT^{2} \)
43 \( 1 + 7.42iT - 43T^{2} \)
47 \( 1 + (-0.928 + 3.46i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 - 3.63iT - 53T^{2} \)
59 \( 1 + (-0.187 + 0.699i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + (0.690 + 0.690i)T + 67iT^{2} \)
71 \( 1 + (1.84 + 6.87i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.38 + 3.38i)T - 73iT^{2} \)
79 \( 1 + (3.69 + 6.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.8 - 3.18i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.512 - 1.91i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.41 - 4.41i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.03397324109907918189085743663, −11.07179936694065348935784046540, −10.46442096218553048608134779417, −9.543232632108439803614986697371, −8.746180394114616896432797094761, −7.56808726116132783377315438885, −6.06668830383665271245798989083, −4.65158292312950449913506690440, −3.49576643217525880573789022461, −2.35646302185415736984197050492, 1.10128160496247438849510158010, 3.13447659633336656521644037699, 5.01561701739341805436251299586, 6.00111073844031544179447247219, 7.15020490882589427728412268428, 8.065752006577822007651953550406, 8.812699695459869726387991215125, 9.772534097351300053382872159902, 11.14781638867457846198232118326, 12.35575588878714697886855304262

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