↑

Properties

Label 2-336-7.4-c3-0-6
Degree $2$
Conductor $336$
Sign $-0.896 - 0.442i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 2.59i)3-s + (3.36 + 5.83i)5-s + (11.2 + 14.7i)7-s + (−4.5 − 7.79i)9-s + (−26.8 + 46.4i)11-s + 1.73·13-s − 20.2·15-s + (23.4 − 40.6i)17-s + (−10.0 − 17.4i)19-s + (−55.1 + 7.10i)21-s + (59.4 + 103. i)23-s + (39.8 − 68.9i)25-s + 27·27-s − 103.·29-s + (−78.7 + 136. i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.301 + 0.521i)5-s + (0.606 + 0.794i)7-s + (−0.166 − 0.288i)9-s + (−0.735 + 1.27i)11-s + 0.0370·13-s − 0.347·15-s + (0.334 − 0.580i)17-s + (−0.121 − 0.210i)19-s + (−0.572 + 0.0738i)21-s + (0.539 + 0.933i)23-s + (0.318 − 0.551i)25-s + 0.192·27-s − 0.663·29-s + (−0.456 + 0.790i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.896 - 0.442i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.896 - 0.442i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.276362920\)
\(L(\frac12)\) \(\approx\) \(1.276362920\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 + (-11.2 - 14.7i)T \)
good5 \( 1 + (-3.36 - 5.83i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (26.8 - 46.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 1.73T + 2.19e3T^{2} \)
17 \( 1 + (-23.4 + 40.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (10.0 + 17.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-59.4 - 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 103.T + 2.43e4T^{2} \)
31 \( 1 + (78.7 - 136. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-18.8 - 32.6i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 287.T + 6.89e4T^{2} \)
43 \( 1 + 504.T + 7.95e4T^{2} \)
47 \( 1 + (110. + 190. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (146. - 253. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (297. - 516. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-132. - 229. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-468. + 811. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 545.T + 3.57e5T^{2} \)
73 \( 1 + (149. - 259. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (470. + 814. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 611.T + 5.71e5T^{2} \)
89 \( 1 + (-588. - 1.01e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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

−11.49045214403456428669855760061, −10.52722535118178500010177603961, −9.797400283725430266647918059321, −8.866992176049649862843829747604, −7.68060426505284538328472539845, −6.69728462829961057181468345159, −5.37937115391498390658861664037, −4.79536699962253149829147153064, −3.13805234784768011719960978313, −1.89947012287706118634109088436, 0.46311419345311015787795418369, 1.69777222590767540111366082230, 3.41504881896100551559714854998, 4.87937074534279839797912424498, 5.72975924385579061320108070569, 6.86028159426918333400858701247, 8.018322887185446754689868752929, 8.556922265265767235458495492564, 9.944020409286679311542920298737, 10.91584681204651010089732072395

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