Properties

Label 2-336-7.2-c3-0-22
Degree $2$
Conductor $336$
Sign $-0.989 + 0.146i$
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 + (7.90 − 13.6i)5-s + (−15.2 + 10.5i)7-s + (−4.5 + 7.79i)9-s + (−15.1 − 26.1i)11-s − 61.6·13-s + 47.4·15-s + (−28.0 − 48.6i)17-s + (−69.7 + 120. i)19-s + (−50.2 − 23.6i)21-s + (−4.07 + 7.05i)23-s + (−62.5 − 108. i)25-s − 27·27-s − 0.217·29-s + (88.2 + 152. i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.707 − 1.22i)5-s + (−0.820 + 0.570i)7-s + (−0.166 + 0.288i)9-s + (−0.414 − 0.718i)11-s − 1.31·13-s + 0.816·15-s + (−0.400 − 0.693i)17-s + (−0.841 + 1.45i)19-s + (−0.522 − 0.245i)21-s + (−0.0369 + 0.0639i)23-s + (−0.500 − 0.866i)25-s − 0.192·27-s − 0.00139·29-s + (0.511 + 0.885i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.989 + 0.146i$
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: \((2,\ 336,\ (\ :3/2),\ -0.989 + 0.146i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1560483599\)
\(L(\frac12)\) \(\approx\) \(0.1560483599\)
\(L(\frac{5}{2})\) 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 \( 1 + (15.2 - 10.5i)T \)
good5 \( 1 + (-7.90 + 13.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (15.1 + 26.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 61.6T + 2.19e3T^{2} \)
17 \( 1 + (28.0 + 48.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (69.7 - 120. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (4.07 - 7.05i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 0.217T + 2.43e4T^{2} \)
31 \( 1 + (-88.2 - 152. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (105. - 182. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 293.T + 6.89e4T^{2} \)
43 \( 1 + 434.T + 7.95e4T^{2} \)
47 \( 1 + (-241. + 418. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (10.2 + 17.7i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (115. + 200. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-419. + 726. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (312. + 540. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 227.T + 3.57e5T^{2} \)
73 \( 1 + (21.5 + 37.2i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (154. - 267. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.23e3T + 5.71e5T^{2} \)
89 \( 1 + (572. - 991. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.68e3T + 9.12e5T^{2} \)
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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

−10.25018191448629903730982079626, −9.798347713202680556522711621889, −8.833879742907643929739639042676, −8.244851714819679027756640098632, −6.64866987418591582904247713024, −5.48128807140860553403000941582, −4.84230665883305189467621165217, −3.30079031362791828306006312644, −2.00762851420218274584539190556, −0.04741685790870965054494493425, 2.17938858645135687486608757462, 2.91090521214840620545377601045, 4.46288248550870161095466522203, 6.04141531607069195598870678038, 6.95213234452487734195190424426, 7.33689832124780190316433399796, 8.851000240586810489717070103658, 10.01154115756769457526927381111, 10.32824614383813910071143196372, 11.50160104360333692246209287221

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