↑

Properties

Label 2-357-21.20-c1-0-7
Degree $2$
Conductor $357$
Sign $-0.993 + 0.110i$
Analytic cond. $2.85065$
Root an. cond. $1.68838$
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  + 1.99i·2-s + (1.72 + 0.160i)3-s − 1.98·4-s − 3.32·5-s + (−0.320 + 3.44i)6-s + (−2.59 + 0.533i)7-s + 0.0357i·8-s + (2.94 + 0.553i)9-s − 6.64i·10-s + 5.68i·11-s + (−3.41 − 0.318i)12-s + 0.505i·13-s + (−1.06 − 5.17i)14-s + (−5.73 − 0.534i)15-s − 4.03·16-s + 17-s + ⋯
L(s)  = 1  + 1.41i·2-s + (0.995 + 0.0926i)3-s − 0.991·4-s − 1.48·5-s + (−0.130 + 1.40i)6-s + (−0.979 + 0.201i)7-s + 0.0126i·8-s + (0.982 + 0.184i)9-s − 2.09i·10-s + 1.71i·11-s + (−0.986 − 0.0918i)12-s + 0.140i·13-s + (−0.284 − 1.38i)14-s + (−1.48 − 0.137i)15-s − 1.00·16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(357\)    =    \(3 \cdot 7 \cdot 17\)
Sign: $-0.993 + 0.110i$
Analytic conductor: \(2.85065\)
Root analytic conductor: \(1.68838\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{357} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 357,\ (\ :1/2),\ -0.993 + 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0630708 - 1.14157i\)
\(L(\frac12)\) \(\approx\) \(0.0630708 - 1.14157i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.160i)T \)
7 \( 1 + (2.59 - 0.533i)T \)
17 \( 1 - T \)
good2 \( 1 - 1.99iT - 2T^{2} \)
5 \( 1 + 3.32T + 5T^{2} \)
11 \( 1 - 5.68iT - 11T^{2} \)
13 \( 1 - 0.505iT - 13T^{2} \)
19 \( 1 + 5.91iT - 19T^{2} \)
23 \( 1 + 2.09iT - 23T^{2} \)
29 \( 1 - 5.99iT - 29T^{2} \)
31 \( 1 - 7.30iT - 31T^{2} \)
37 \( 1 - 4.58T + 37T^{2} \)
41 \( 1 - 6.20T + 41T^{2} \)
43 \( 1 - 3.09T + 43T^{2} \)
47 \( 1 + 3.07T + 47T^{2} \)
53 \( 1 - 6.51iT - 53T^{2} \)
59 \( 1 - 1.46T + 59T^{2} \)
61 \( 1 + 4.12iT - 61T^{2} \)
67 \( 1 - 8.33T + 67T^{2} \)
71 \( 1 + 2.85iT - 71T^{2} \)
73 \( 1 - 7.36iT - 73T^{2} \)
79 \( 1 - 9.35T + 79T^{2} \)
83 \( 1 - 3.24T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 - 15.3iT - 97T^{2} \)
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

−12.31081281139671134470381770660, −10.92655144427528531489112926635, −9.600053730314969144822555927588, −8.889355611957531990641852212114, −7.972738610522033209554000027713, −7.11526100319878267038479180310, −6.82203239353185934433410000957, −4.92768638275006956567731138046, −4.11626461410495539371949274985, −2.74495838338562964502074147881, 0.71038574107784071302859733670, 2.71715494519737771506064197945, 3.68171505640365103292303044973, 3.93947226284972514559612052603, 6.19031834691244288355953875643, 7.60556934087446920987819952055, 8.261880518795548399823957610465, 9.335586966965691258577318094285, 10.12308568725841984176819033357, 11.13296913446614731978041943786

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