↑

Properties

Label 2-3696-3696.2309-c0-0-11
Degree $2$
Conductor $3696$
Sign $-0.608 - 0.793i$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)2-s + (0.707 − 0.707i)3-s + (−0.866 + 0.499i)4-s + (−1.22 − 1.22i)5-s + (−0.866 − 0.500i)6-s − i·7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (−0.866 + 1.49i)10-s + (−0.707 − 0.707i)11-s + (−0.258 + 0.965i)12-s + (1.36 − 1.36i)13-s + (−0.965 + 0.258i)14-s − 1.73·15-s + (0.500 − 0.866i)16-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (0.707 − 0.707i)3-s + (−0.866 + 0.499i)4-s + (−1.22 − 1.22i)5-s + (−0.866 − 0.500i)6-s − i·7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (−0.866 + 1.49i)10-s + (−0.707 − 0.707i)11-s + (−0.258 + 0.965i)12-s + (1.36 − 1.36i)13-s + (−0.965 + 0.258i)14-s − 1.73·15-s + (0.500 − 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3696\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3696} (2309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3696,\ (\ :0),\ -0.608 - 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.051559785\)
\(L(\frac12)\) \(\approx\) \(1.051559785\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + iT \)
11 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
13 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - 0.517T + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
61 \( 1 + (-1 + i)T - iT^{2} \)
67 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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

−8.193029013329683366641998622377, −8.012133628816024081538280501721, −7.19663972431673373789765085513, −5.89815788310178981069372215130, −4.89520937324516421477793617136, −4.01769448120342718232970769629, −3.49058983323003355798914932604, −2.72335335598458024639173857830, −0.962896054150580955874786179650, −0.867286802115551056863611995596, 2.03918043537119474410717803655, 3.10670844921963810937798854824, 3.93509394091055834066027805216, 4.49255679238178345182900978884, 5.51153811368959215855983160773, 6.38277302206027970498122855950, 7.13034539098367856763026489000, 7.74913186437334868506603397811, 8.400267376887529667233773186582, 8.943947928886193126032356534389

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