↑

Properties

Label 2-336-112.101-c2-0-36
Degree $2$
Conductor $336$
Sign $0.993 + 0.116i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.99 + 0.0450i)2-s + (1.67 − 0.448i)3-s + (3.99 − 0.180i)4-s + (5.27 + 1.41i)5-s + (−3.32 + 0.971i)6-s + (6.64 − 2.20i)7-s + (−7.98 + 0.540i)8-s + (2.59 − 1.50i)9-s + (−10.6 − 2.58i)10-s + (11.6 − 3.12i)11-s + (6.60 − 2.09i)12-s + (7.75 + 7.75i)13-s + (−13.1 + 4.71i)14-s + 9.45·15-s + (15.9 − 1.44i)16-s + (−16.8 − 9.70i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0225i)2-s + (0.557 − 0.149i)3-s + (0.998 − 0.0450i)4-s + (1.05 + 0.282i)5-s + (−0.554 + 0.161i)6-s + (0.948 − 0.315i)7-s + (−0.997 + 0.0675i)8-s + (0.288 − 0.166i)9-s + (−1.06 − 0.258i)10-s + (1.06 − 0.284i)11-s + (0.550 − 0.174i)12-s + (0.596 + 0.596i)13-s + (−0.941 + 0.336i)14-s + 0.630·15-s + (0.995 − 0.0900i)16-s + (−0.988 − 0.570i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.993 + 0.116i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.993 + 0.116i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.79402 - 0.104718i\)
\(L(\frac12)\) \(\approx\) \(1.79402 - 0.104718i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 - 0.0450i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
7 \( 1 + (-6.64 + 2.20i)T \)
good5 \( 1 + (-5.27 - 1.41i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-11.6 + 3.12i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-7.75 - 7.75i)T + 169iT^{2} \)
17 \( 1 + (16.8 + 9.70i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-0.915 + 3.41i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (26.0 - 15.0i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (24.5 - 24.5i)T - 841iT^{2} \)
31 \( 1 + (-16.1 - 9.32i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-15.9 + 59.7i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 1.84T + 1.68e3T^{2} \)
43 \( 1 + (-13.4 - 13.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (3.95 - 2.28i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-49.5 + 13.2i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-20.4 - 76.3i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (2.21 - 8.26i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (1.88 + 7.05i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 57.3iT - 5.04e3T^{2} \)
73 \( 1 + (32.0 - 55.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-5.75 - 9.96i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (58.1 + 58.1i)T + 6.88e3iT^{2} \)
89 \( 1 + (68.3 + 118. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 37.9iT - 9.40e3T^{2} \)
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

−11.18938330514981440587786397115, −10.24955471576291201028123527219, −9.208323013950913437255998832518, −8.812601400297655238230271854513, −7.58801181277302556847122969310, −6.71800921358106578754072332013, −5.76815205264316057205738219022, −3.97033410516910228350517495373, −2.28765080126880346156425994062, −1.40056428647588453349317294545, 1.46767135341269982144257550441, 2.32815495272863702038075088438, 4.12111951895193524240882547948, 5.72494627093188262220139453840, 6.56625642323177985386105751107, 8.002222689035335607905456863317, 8.574632459529438310960953057572, 9.449061349553156246374582762149, 10.15110625955132196314880429164, 11.16179366742914083179012905861

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