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Properties

Label 2-336-28.19-c3-0-22
Degree $2$
Conductor $336$
Sign $-0.978 + 0.205i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 2.59i)3-s + (−1.5 − 0.866i)5-s + (3.5 − 18.1i)7-s + (−4.5 + 7.79i)9-s + (−34.5 + 19.9i)11-s − 34.6i·13-s − 5.19i·15-s + (−81 + 46.7i)17-s + (−47 + 81.4i)19-s + (52.5 − 18.1i)21-s + (78 + 45.0i)23-s + (−61 − 105. i)25-s − 27·27-s − 225·29-s + (−5.5 − 9.52i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.134 − 0.0774i)5-s + (0.188 − 0.981i)7-s + (−0.166 + 0.288i)9-s + (−0.945 + 0.545i)11-s − 0.739i·13-s − 0.0894i·15-s + (−1.15 + 0.667i)17-s + (−0.567 + 0.982i)19-s + (0.545 − 0.188i)21-s + (0.707 + 0.408i)23-s + (−0.487 − 0.845i)25-s − 0.192·27-s − 1.44·29-s + (−0.0318 − 0.0551i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.978 + 0.205i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.978 + 0.205i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.5 + 18.1i)T \)
good5 \( 1 + (1.5 + 0.866i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (34.5 - 19.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 34.6iT - 2.19e3T^{2} \)
17 \( 1 + (81 - 46.7i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (47 - 81.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-78 - 45.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 225T + 2.43e4T^{2} \)
31 \( 1 + (5.5 + 9.52i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (179 - 310. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 142. iT - 6.89e4T^{2} \)
43 \( 1 + 398. iT - 7.95e4T^{2} \)
47 \( 1 + (-96 + 166. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (208.5 + 361. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-280.5 - 485. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (462 + 266. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-84 + 48.4i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 838. iT - 3.57e5T^{2} \)
73 \( 1 + (18 - 10.3i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (16.5 + 9.52i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 693T + 5.71e5T^{2} \)
89 \( 1 + (885 + 510. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 822. iT - 9.12e5T^{2} \)
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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

−10.42816820060875177209556161826, −10.11350241300699496559686047862, −8.703059438029110940706129276916, −7.924667828614949123166468957021, −7.00151136400332508628681726683, −5.57834627042492679058005713078, −4.47640060092204602627770239979, −3.55676014326945022677220083062, −1.98863001593976343892261436119, 0, 2.01600139395742909364106300538, 2.97403333323051362084706578867, 4.62703483952176866488821527689, 5.73294588577397202555474505373, 6.83213915758802382845440843753, 7.77381384905319206279790000833, 8.892722147642145279803686290010, 9.293602885989573499760974839062, 11.07277452345220138171865353375

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