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Properties

Label 2-336-21.5-c1-0-5
Degree $2$
Conductor $336$
Sign $0.744 - 0.667i$
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  + (−1.5 + 0.866i)3-s + (2.5 − 0.866i)7-s + (1.5 − 2.59i)9-s + 5.19i·13-s + (7.5 + 4.33i)19-s + (−3 + 3.46i)21-s + (2.5 + 4.33i)25-s + 5.19i·27-s + (1.5 − 0.866i)31-s + (5.5 − 9.52i)37-s + (−4.5 − 7.79i)39-s − 13·43-s + (5.5 − 4.33i)49-s − 15·57-s + (−6 − 3.46i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (0.944 − 0.327i)7-s + (0.5 − 0.866i)9-s + 1.44i·13-s + (1.72 + 0.993i)19-s + (−0.654 + 0.755i)21-s + (0.5 + 0.866i)25-s + 0.999i·27-s + (0.269 − 0.155i)31-s + (0.904 − 1.56i)37-s + (−0.720 − 1.24i)39-s − 1.98·43-s + (0.785 − 0.618i)49-s − 1.98·57-s + (−0.768 − 0.443i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.744 - 0.667i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05330 + 0.402686i\)
\(L(\frac12)\) \(\approx\) \(1.05330 + 0.402686i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.5 - 4.33i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 13T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6 + 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (1.5 - 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.5 + 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−11.58244241207607672689151658780, −10.92233883042972464672070943848, −9.862873623354153546109094812114, −9.097687152181095468843368427585, −7.74182725700019205083396136380, −6.82262510471165382049645141676, −5.60303263339229177656622158027, −4.72489118835407593471470249125, −3.68832397018173474483259130812, −1.47069390682048601460837366206, 1.08663811215078080285330161752, 2.83556710008285013557932878601, 4.81830220351592587454175174121, 5.38936203726350020580573736052, 6.57571978741378399172294032276, 7.68126515690078702902154452946, 8.344963420014972279445098398002, 9.801622961920354228484043764735, 10.70866196612675286434898990053, 11.55433317339678327785464475661

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