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Properties

Label 2-336-7.2-c5-0-17
Degree $2$
Conductor $336$
Sign $0.874 + 0.485i$
Analytic cond. $53.8889$
Root an. cond. $7.34091$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 − 7.79i)3-s + (−5.5 + 9.52i)5-s + (−129.5 − 6.06i)7-s + (−40.5 + 70.1i)9-s + (134.5 + 232. i)11-s − 308·13-s + 99·15-s + (−948 − 1.64e3i)17-s + (−82 + 142. i)19-s + (535.5 + 1.03e3i)21-s + (−1.63e3 + 2.82e3i)23-s + (1.50e3 + 2.60e3i)25-s + 729·27-s + 2.41e3·29-s + (1.42e3 + 2.46e3i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.0983 + 0.170i)5-s + (−0.998 − 0.0467i)7-s + (−0.166 + 0.288i)9-s + (0.335 + 0.580i)11-s − 0.505·13-s + 0.113·15-s + (−0.795 − 1.37i)17-s + (−0.0521 + 0.0902i)19-s + (0.264 + 0.512i)21-s + (−0.643 + 1.11i)23-s + (0.480 + 0.832i)25-s + 0.192·27-s + 0.533·29-s + (0.265 + 0.459i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.874 + 0.485i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ 0.874 + 0.485i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.165286173\)
\(L(\frac12)\) \(\approx\) \(1.165286173\)
\(L(\frac{7}{2})\) 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 \( 1 + (4.5 + 7.79i)T \)
7 \( 1 + (129.5 + 6.06i)T \)
good5 \( 1 + (5.5 - 9.52i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-134.5 - 232. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 308T + 3.71e5T^{2} \)
17 \( 1 + (948 + 1.64e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (82 - 142. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.63e3 - 2.82e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 2.41e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.42e3 - 2.46e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.66e3 + 9.81e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.68e4T + 1.15e8T^{2} \)
43 \( 1 - 7.89e3T + 1.47e8T^{2} \)
47 \( 1 + (-1.05e4 + 1.82e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.48e4 - 2.57e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (4.08e3 + 7.06e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (7.58e3 - 1.31e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.60e4 + 2.77e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 3.82e4T + 1.80e9T^{2} \)
73 \( 1 + (1.74e4 + 3.01e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-6.76e3 + 1.17e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 6.81e4T + 3.93e9T^{2} \)
89 \( 1 + (-5.74e4 + 9.95e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.54e5T + 8.58e9T^{2} \)
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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

−10.65827841284478604070829963581, −9.670769930371010551511434840483, −8.947188659693247355317874545494, −7.39762362155099180753867792678, −6.98597622819448224138292409434, −5.88343009600636070976599811557, −4.70603916143186200625181129795, −3.32837487493834176531179457756, −2.13110917256450655504554391954, −0.54057955064888260603288394228, 0.62687276675381921739680404473, 2.50032862763346984260423707938, 3.76877995862638132193220314942, 4.68699207263619372441088388012, 6.13296319990750587653923063709, 6.58453873755212944780955099214, 8.200731726028440160812814511422, 8.944471816907030812163915303212, 10.03454532311051996597712642485, 10.59313111766325320931401261642

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