↑

Properties

Label 2-234-117.11-c1-0-7
Degree $2$
Conductor $234$
Sign $0.938 - 0.345i$
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 + (1.67 − 0.444i)3-s − 1.00i·4-s + (3.83 + 1.02i)5-s + (−0.869 + 1.49i)6-s + (−1.55 − 0.415i)7-s + (0.707 + 0.707i)8-s + (2.60 − 1.48i)9-s + (−3.43 + 1.98i)10-s + (−3.50 − 3.50i)11-s + (−0.444 − 1.67i)12-s + (−1.03 + 3.45i)13-s + (1.39 − 0.803i)14-s + (6.87 + 0.0145i)15-s − 1.00·16-s + (0.584 − 1.01i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.966 − 0.256i)3-s − 0.500i·4-s + (1.71 + 0.459i)5-s + (−0.354 + 0.611i)6-s + (−0.586 − 0.157i)7-s + (0.250 + 0.250i)8-s + (0.868 − 0.496i)9-s + (−1.08 + 0.627i)10-s + (−1.05 − 1.05i)11-s + (−0.128 − 0.483i)12-s + (−0.288 + 0.957i)13-s + (0.371 − 0.214i)14-s + (1.77 + 0.00375i)15-s − 0.250·16-s + (0.141 − 0.245i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.938 - 0.345i$
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: \((2,\ 234,\ (\ :1/2),\ 0.938 - 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48255 + 0.264355i\)
\(L(\frac12)\) \(\approx\) \(1.48255 + 0.264355i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 + 0.444i)T \)
13 \( 1 + (1.03 - 3.45i)T \)
good5 \( 1 + (-3.83 - 1.02i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.55 + 0.415i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.50 + 3.50i)T + 11iT^{2} \)
17 \( 1 + (-0.584 + 1.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.16 - 1.11i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.63 + 2.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.50iT - 29T^{2} \)
31 \( 1 + (1.94 - 7.25i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.28 + 1.14i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.44 + 5.41i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.770 - 0.444i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.43 + 0.653i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 6.60iT - 53T^{2} \)
59 \( 1 + (-5.81 - 5.81i)T + 59iT^{2} \)
61 \( 1 + (7.05 + 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.19 - 0.855i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.942 - 3.51i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.33 - 3.33i)T - 73iT^{2} \)
79 \( 1 + (1.16 - 2.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.82 + 10.5i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (1.78 - 6.65i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.65 - 6.17i)T + (-84.0 - 48.5i)T^{2} \)
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

−12.66829401593755248949645317042, −10.68117167944282690393444898131, −10.17267753568737865203088541055, −9.140805977313498920091908534310, −8.568454091458252745490078295210, −7.05621346328667778594614068241, −6.45940698429993905504441488995, −5.24648075824000802976927669520, −3.11197061608121771681908992683, −1.92020018230108385184546902031, 1.98930521193740247269392832466, 2.81439928967093052849388242729, 4.64737747586945621350229126505, 5.90974694288548258249387328447, 7.42597538419669795384917964109, 8.472494198252959903913748272585, 9.551592097861042272063430286599, 9.893900900267100323883326787525, 10.61484545842654447147265775755, 12.52239603647713144419858402923

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