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Properties

Label 2-336-336.5-c1-0-19
Degree $2$
Conductor $336$
Sign $0.433 - 0.901i$
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  + (0.590 + 1.28i)2-s + (−1.32 − 1.11i)3-s + (−1.30 + 1.51i)4-s + (0.973 − 0.260i)5-s + (0.650 − 2.36i)6-s + (2.63 − 0.251i)7-s + (−2.71 − 0.777i)8-s + (0.512 + 2.95i)9-s + (0.909 + 1.09i)10-s + (0.460 + 0.123i)11-s + (3.41 − 0.558i)12-s + (2.21 + 2.21i)13-s + (1.87 + 3.23i)14-s + (−1.58 − 0.739i)15-s + (−0.606 − 3.95i)16-s + (2.95 + 5.11i)17-s + ⋯
L(s)  = 1  + (0.417 + 0.908i)2-s + (−0.765 − 0.643i)3-s + (−0.651 + 0.758i)4-s + (0.435 − 0.116i)5-s + (0.265 − 0.964i)6-s + (0.995 − 0.0950i)7-s + (−0.961 − 0.274i)8-s + (0.170 + 0.985i)9-s + (0.287 + 0.346i)10-s + (0.138 + 0.0371i)11-s + (0.986 − 0.161i)12-s + (0.613 + 0.613i)13-s + (0.502 + 0.864i)14-s + (−0.408 − 0.191i)15-s + (−0.151 − 0.988i)16-s + (0.716 + 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.433 - 0.901i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.433 - 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20915 + 0.760435i\)
\(L(\frac12)\) \(\approx\) \(1.20915 + 0.760435i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.590 - 1.28i)T \)
3 \( 1 + (1.32 + 1.11i)T \)
7 \( 1 + (-2.63 + 0.251i)T \)
good5 \( 1 + (-0.973 + 0.260i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.460 - 0.123i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.21 - 2.21i)T + 13iT^{2} \)
17 \( 1 + (-2.95 - 5.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.22 + 1.66i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.412 - 0.714i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.21 + 1.21i)T + 29iT^{2} \)
31 \( 1 + (5.94 - 3.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.25 + 0.872i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.00iT - 41T^{2} \)
43 \( 1 + (5.59 + 5.59i)T + 43iT^{2} \)
47 \( 1 + (-5.11 + 8.86i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.68 + 1.25i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.46 + 12.9i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.90 + 0.511i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (11.2 + 3.00i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 + (2.26 + 3.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.39 - 14.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.39 - 9.39i)T - 83iT^{2} \)
89 \( 1 + (5.39 + 3.11i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.65iT - 97T^{2} \)
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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

−11.82611318527828290809971684177, −11.11186013419795201460928686696, −9.778899146014722851426268436904, −8.553679501655085351447491420939, −7.71097456215372368453781807311, −6.86090644872717428212892049560, −5.73831108814099435795974898273, −5.19764408715442969823119049456, −3.84514593271398871263653907434, −1.60290904175554104131567442677, 1.21382076619336619109882749856, 3.06062725434308644320036364844, 4.31358843807088971078168989587, 5.39157102497901312460549578448, 5.88116628939512989991183770362, 7.61306130440657359994511193207, 9.053141220114841196370475565548, 9.822363742523822874350470589049, 10.59675504589377242208204135586, 11.53809117085546444308783183410

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