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Properties

Label 2-336-7.2-c3-0-20
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

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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: $\chi_{336} (289, \cdot )$
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 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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