↑

Properties

Label 2-234-117.11-c1-0-8
Degree $2$
Conductor $234$
Sign $0.214 + 0.976i$
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  + (0.707 − 0.707i)2-s + (−0.539 − 1.64i)3-s − 1.00i·4-s + (3.04 + 0.817i)5-s + (−1.54 − 0.781i)6-s + (2.84 + 0.761i)7-s + (−0.707 − 0.707i)8-s + (−2.41 + 1.77i)9-s + (2.73 − 1.57i)10-s + (−0.616 − 0.616i)11-s + (−1.64 + 0.539i)12-s + (−3.08 − 1.86i)13-s + (2.54 − 1.47i)14-s + (−0.301 − 5.45i)15-s − 1.00·16-s + (−1.80 + 3.11i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.311 − 0.950i)3-s − 0.500i·4-s + (1.36 + 0.365i)5-s + (−0.630 − 0.319i)6-s + (1.07 + 0.287i)7-s + (−0.250 − 0.250i)8-s + (−0.805 + 0.592i)9-s + (0.864 − 0.499i)10-s + (−0.185 − 0.185i)11-s + (−0.475 + 0.155i)12-s + (−0.855 − 0.517i)13-s + (0.681 − 0.393i)14-s + (−0.0779 − 1.40i)15-s − 0.250·16-s + (−0.436 + 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.214 + 0.976i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.214 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35979 - 1.09315i\)
\(L(\frac12)\) \(\approx\) \(1.35979 - 1.09315i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.539 + 1.64i)T \)
13 \( 1 + (3.08 + 1.86i)T \)
good5 \( 1 + (-3.04 - 0.817i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.84 - 0.761i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.616 + 0.616i)T + 11iT^{2} \)
17 \( 1 + (1.80 - 3.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.90 - 1.31i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.86 + 3.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.59iT - 29T^{2} \)
31 \( 1 + (-1.33 + 4.96i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (5.77 + 1.54i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.82 - 10.5i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.48 + 3.16i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.10 + 1.63i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 6.73iT - 53T^{2} \)
59 \( 1 + (3.89 + 3.89i)T + 59iT^{2} \)
61 \( 1 + (3.18 + 5.51i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.99 - 2.14i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.65 + 6.19i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-10.6 + 10.6i)T - 73iT^{2} \)
79 \( 1 + (1.30 - 2.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.96 - 11.0i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-2.10 + 7.84i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.37 - 16.3i)T + (-84.0 - 48.5i)T^{2} \)
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

−12.22792732146792045695025667891, −10.86352981788142585525049086243, −10.57563790095798019752726276539, −9.075173008670494226460023339957, −7.956942369715327797262414112027, −6.60078136649432439358627436650, −5.76426011015337419441678555328, −4.84884714509496938914410382114, −2.58719998429962459162531902816, −1.73660209323026482743477279687, 2.32705024925811297536814991884, 4.37252847207909906343890767438, 5.02568661355280824123589429612, 5.93776891768198565088241423186, 7.19386216674122025712352325849, 8.661858524750460977471696250755, 9.450182168605179404831457886954, 10.42807411469472661708284037998, 11.39813453206621867688886799395, 12.40564105162204584446888190964

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