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Properties

Label 2-234-117.103-c1-0-10
Degree $2$
Conductor $234$
Sign $0.365 + 0.930i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.215 − 1.71i)3-s + (0.499 − 0.866i)4-s + (2.40 + 1.38i)5-s + (−1.04 − 1.38i)6-s + (0.759 − 0.438i)7-s − 0.999i·8-s + (−2.90 + 0.739i)9-s + 2.77·10-s + (1.92 − 1.11i)11-s + (−1.59 − 0.672i)12-s + (−3.20 + 1.64i)13-s + (0.438 − 0.759i)14-s + (1.86 − 4.43i)15-s + (−0.5 − 0.866i)16-s − 3.80·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.124 − 0.992i)3-s + (0.249 − 0.433i)4-s + (1.07 + 0.621i)5-s + (−0.426 − 0.563i)6-s + (0.286 − 0.165i)7-s − 0.353i·8-s + (−0.969 + 0.246i)9-s + 0.878·10-s + (0.581 − 0.335i)11-s + (−0.460 − 0.194i)12-s + (−0.889 + 0.456i)13-s + (0.117 − 0.202i)14-s + (0.482 − 1.14i)15-s + (−0.125 − 0.216i)16-s − 0.923·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.365 + 0.930i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.365 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52631 - 1.04093i\)
\(L(\frac12)\) \(\approx\) \(1.52631 - 1.04093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.215 + 1.71i)T \)
13 \( 1 + (3.20 - 1.64i)T \)
good5 \( 1 + (-2.40 - 1.38i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.759 + 0.438i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.92 + 1.11i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 3.80T + 17T^{2} \)
19 \( 1 - 2.22iT - 19T^{2} \)
23 \( 1 + (-0.259 + 0.449i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.81 + 6.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.97 - 2.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 11.3iT - 37T^{2} \)
41 \( 1 + (-3.52 - 2.03i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.81 - 4.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.920 - 0.531i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.29T + 53T^{2} \)
59 \( 1 + (3.52 + 2.03i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.94 + 6.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.95 + 3.43i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.9iT - 71T^{2} \)
73 \( 1 - 5.24iT - 73T^{2} \)
79 \( 1 + (7.09 + 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.641 + 0.370i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.89iT - 89T^{2} \)
97 \( 1 + (-13.6 + 7.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.98419222986432052556116013570, −11.30724967352228256760630761007, −10.25397358715709947359172304521, −9.225120491998427459727987205377, −7.79399618388661620345689482052, −6.55404338909434768484874293131, −6.10288608431691434783171903640, −4.67090983846081450916124320396, −2.80933503311809719229708038174, −1.73249364984201211515764588036, 2.40383319136938177395486099518, 4.13737105311704984014047091355, 5.12377259421575433068951342532, 5.81258006090283039034549864465, 7.15992475684635156366870668845, 8.762254898433017525833121523903, 9.345631233391249159283685707441, 10.39549564080408204979544257161, 11.44630189111850801195664244409, 12.46958562878454283180784461315

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