↑

Properties

Label 2-2646-21.20-c1-0-41
Degree $2$
Conductor $2646$
Sign $0.987 - 0.156i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
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  + i·2-s − 4-s + 3.21·5-s − i·8-s + 3.21i·10-s + 0.674i·11-s − 0.881i·13-s + 16-s + 7.27·17-s − 7.76i·19-s − 3.21·20-s − 0.674·22-s − 4.35i·23-s + 5.33·25-s + 0.881·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.43·5-s − 0.353i·8-s + 1.01i·10-s + 0.203i·11-s − 0.244i·13-s + 0.250·16-s + 1.76·17-s − 1.78i·19-s − 0.718·20-s − 0.143·22-s − 0.907i·23-s + 1.06·25-s + 0.172·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.987 - 0.156i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.987 - 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.330476745\)
\(L(\frac12)\) \(\approx\) \(2.330476745\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 3.21T + 5T^{2} \)
11 \( 1 - 0.674iT - 11T^{2} \)
13 \( 1 + 0.881iT - 13T^{2} \)
17 \( 1 - 7.27T + 17T^{2} \)
19 \( 1 + 7.76iT - 19T^{2} \)
23 \( 1 + 4.35iT - 23T^{2} \)
29 \( 1 + 5.15iT - 29T^{2} \)
31 \( 1 + 2.66iT - 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + 6.36T + 41T^{2} \)
43 \( 1 - 9.96T + 43T^{2} \)
47 \( 1 - 3.28T + 47T^{2} \)
53 \( 1 + 2.60iT - 53T^{2} \)
59 \( 1 + 5.75T + 59T^{2} \)
61 \( 1 - 6.98iT - 61T^{2} \)
67 \( 1 - 7.94T + 67T^{2} \)
71 \( 1 + 4.11iT - 71T^{2} \)
73 \( 1 - 13.8iT - 73T^{2} \)
79 \( 1 + 1.21T + 79T^{2} \)
83 \( 1 - 1.71T + 83T^{2} \)
89 \( 1 - 6.94T + 89T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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

−8.922705075897342664154817559653, −8.111102325801115508040949090730, −7.20408840175657270715821423479, −6.58149967253850788824647920806, −5.70150516763824781327336538097, −5.30428777872165741920706203440, −4.35665887950704315487041164781, −3.08618556321154895067131530663, −2.15418553149921814339146378194, −0.827683382890789604134482773524, 1.33394808517205169816291375013, 1.85236720427055038110076692805, 3.13627373861879619023920879383, 3.73683329240861490997839758984, 5.17809309246662566909258573796, 5.55763257374339295756542521507, 6.31114240032869059383213961900, 7.41557448243667775258905312060, 8.216898799761582873668988064290, 9.143342290592252385941632873340

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