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Properties

Label 2-336-112.109-c1-0-24
Degree $2$
Conductor $336$
Sign $-0.613 + 0.789i$
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  + (−0.687 − 1.23i)2-s + (0.965 − 0.258i)3-s + (−1.05 + 1.69i)4-s + (0.137 + 0.0368i)5-s + (−0.983 − 1.01i)6-s + (−2.08 − 1.62i)7-s + (2.82 + 0.137i)8-s + (0.866 − 0.499i)9-s + (−0.0490 − 0.195i)10-s + (−1.29 − 4.82i)11-s + (−0.580 + 1.91i)12-s + (−0.0957 − 0.0957i)13-s + (−0.579 + 3.69i)14-s + 0.142·15-s + (−1.77 − 3.58i)16-s + (2.58 − 4.48i)17-s + ⋯
L(s)  = 1  + (−0.485 − 0.874i)2-s + (0.557 − 0.149i)3-s + (−0.527 + 0.849i)4-s + (0.0615 + 0.0165i)5-s + (−0.401 − 0.414i)6-s + (−0.788 − 0.615i)7-s + (0.998 + 0.0487i)8-s + (0.288 − 0.166i)9-s + (−0.0154 − 0.0618i)10-s + (−0.389 − 1.45i)11-s + (−0.167 + 0.552i)12-s + (−0.0265 − 0.0265i)13-s + (−0.154 + 0.987i)14-s + 0.0368·15-s + (−0.442 − 0.896i)16-s + (0.628 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.613 + 0.789i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.613 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.443690 - 0.906283i\)
\(L(\frac12)\) \(\approx\) \(0.443690 - 0.906283i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.687 + 1.23i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (2.08 + 1.62i)T \)
good5 \( 1 + (-0.137 - 0.0368i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.29 + 4.82i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.0957 + 0.0957i)T + 13iT^{2} \)
17 \( 1 + (-2.58 + 4.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.638 + 2.38i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.16 + 2.40i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.22 - 6.22i)T + 29iT^{2} \)
31 \( 1 + (2.44 - 4.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.01 + 1.34i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.89iT - 41T^{2} \)
43 \( 1 + (0.390 - 0.390i)T - 43iT^{2} \)
47 \( 1 + (-1.18 - 2.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.658 - 2.45i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.09 + 7.81i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.17 - 4.39i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.284 + 0.0763i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.82iT - 71T^{2} \)
73 \( 1 + (-12.3 - 7.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.01 - 8.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.66 + 1.66i)T + 83iT^{2} \)
89 \( 1 + (-10.5 + 6.08i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−11.03077272683548972774359172574, −10.33978712129122452577048755586, −9.380567574301816530021376969183, −8.635116821309847447789691465113, −7.65314399192257570145950992546, −6.65549547405307893428979435600, −4.97901984879177543171407978108, −3.44728808245592431567106019656, −2.84736273062114643438025830501, −0.810714921574673092857728627531, 2.02452667266758360003346665759, 3.80548562900145797336034404637, 5.15894745439811014551937518322, 6.16738739710268959625595661446, 7.27692727411721875258741261735, 8.055455139157785913415157610454, 9.111519560086143876600829771279, 9.832885923855854225259283441555, 10.40064139295408063903952962775, 12.04659012749247420593696451092

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