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Properties

Label 2-336-84.11-c1-0-7
Degree $2$
Conductor $336$
Sign $0.843 - 0.536i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.906 + 1.47i)3-s + (3.41 − 1.97i)5-s + (−1.13 + 2.38i)7-s + (−1.35 + 2.67i)9-s + (1.76 − 3.05i)11-s + 2·13-s + (6.00 + 3.25i)15-s + (−4.35 − 2.51i)17-s + (−2.63 + 1.52i)19-s + (−4.55 + 0.485i)21-s + (0.825 + 1.42i)23-s + (5.27 − 9.13i)25-s + (−5.17 + 0.418i)27-s + 6.80i·29-s + (1.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (0.523 + 0.852i)3-s + (1.52 − 0.881i)5-s + (−0.429 + 0.902i)7-s + (−0.452 + 0.891i)9-s + (0.531 − 0.921i)11-s + 0.554·13-s + (1.55 + 0.840i)15-s + (−1.05 − 0.609i)17-s + (−0.605 + 0.349i)19-s + (−0.994 + 0.105i)21-s + (0.172 + 0.298i)23-s + (1.05 − 1.82i)25-s + (−0.996 + 0.0805i)27-s + 1.26i·29-s + (0.269 + 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.843 - 0.536i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.843 - 0.536i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78460 + 0.519264i\)
\(L(\frac12)\) \(\approx\) \(1.78460 + 0.519264i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.906 - 1.47i)T \)
7 \( 1 + (1.13 - 2.38i)T \)
good5 \( 1 + (-3.41 + 1.97i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.76 + 3.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (4.35 + 2.51i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.63 - 1.52i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.825 - 1.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.80iT - 29T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.637 + 1.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.16iT - 41T^{2} \)
43 \( 1 + 0.837iT - 43T^{2} \)
47 \( 1 + (2.47 + 4.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.36 + 4.83i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.06 + 8.77i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.63 - 9.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.9 + 8.03i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + (-3.63 + 6.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.17T + 83T^{2} \)
89 \( 1 + (6.23 - 3.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.27T + 97T^{2} \)
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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

−11.51642669905747471410728276199, −10.49770052081469235287675898929, −9.540696617271595604171739959924, −8.894330571023533567062351902497, −8.541556153203488186247230434897, −6.45600035249162833172386614749, −5.65191017646482527661982486491, −4.76235680191821832389416154387, −3.22427131049035151384514393902, −1.93332311410992284739918870589, 1.66965200415206305592440109853, 2.71940577654385166953590424763, 4.18625905651917666569004456217, 6.16757466618971776165312852581, 6.54187270463832897068087593238, 7.39398504467975340194476220286, 8.774289721925042075252831830852, 9.658891072749538303230504522293, 10.40542942765406734552936885006, 11.36807104685916879794480383377

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