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Properties

Label 2-336-112.101-c2-0-14
Degree $2$
Conductor $336$
Sign $0.389 - 0.920i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.823 + 1.82i)2-s + (1.67 − 0.448i)3-s + (−2.64 − 3.00i)4-s + (−9.06 − 2.42i)5-s + (−0.559 + 3.41i)6-s + (−6.77 + 1.75i)7-s + (7.64 − 2.35i)8-s + (2.59 − 1.50i)9-s + (11.8 − 14.5i)10-s + (15.0 − 4.03i)11-s + (−5.77 − 3.83i)12-s + (8.68 + 8.68i)13-s + (2.38 − 13.7i)14-s − 16.2·15-s + (−2.00 + 15.8i)16-s + (12.8 + 7.40i)17-s + ⋯
L(s)  = 1  + (−0.411 + 0.911i)2-s + (0.557 − 0.149i)3-s + (−0.661 − 0.750i)4-s + (−1.81 − 0.485i)5-s + (−0.0933 + 0.569i)6-s + (−0.968 + 0.250i)7-s + (0.955 − 0.294i)8-s + (0.288 − 0.166i)9-s + (1.18 − 1.45i)10-s + (1.36 − 0.366i)11-s + (−0.480 − 0.319i)12-s + (0.668 + 0.668i)13-s + (0.170 − 0.985i)14-s − 1.08·15-s + (−0.125 + 0.992i)16-s + (0.754 + 0.435i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.389 - 0.920i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.389 - 0.920i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.828864 + 0.549292i\)
\(L(\frac12)\) \(\approx\) \(0.828864 + 0.549292i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.823 - 1.82i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
7 \( 1 + (6.77 - 1.75i)T \)
good5 \( 1 + (9.06 + 2.42i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-15.0 + 4.03i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-8.68 - 8.68i)T + 169iT^{2} \)
17 \( 1 + (-12.8 - 7.40i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (4.71 - 17.5i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (8.25 - 4.76i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-28.7 + 28.7i)T - 841iT^{2} \)
31 \( 1 + (3.46 + 2.00i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-0.253 + 0.945i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 23.1T + 1.68e3T^{2} \)
43 \( 1 + (-21.1 - 21.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-72.7 + 42.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-20.9 + 5.61i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-4.88 - 18.2i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (28.6 - 106. i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (2.12 + 7.92i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 22.4iT - 5.04e3T^{2} \)
73 \( 1 + (16.6 - 28.8i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-10.9 - 19.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-98.1 - 98.1i)T + 6.88e3iT^{2} \)
89 \( 1 + (46.7 + 80.9i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 114. iT - 9.40e3T^{2} \)
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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

−11.74858079169463310954297029988, −10.33833510281285442995165208398, −9.168828067533411482634112336265, −8.603771909764415061140536644460, −7.84355834326771570913202826960, −6.86606669132526182915391705645, −5.96545384467251726208138137029, −4.13285161966756216529022020299, −3.71236688037677485028937081229, −0.994098477920903070641961072948, 0.71957270738804711339005990949, 3.00818855057367088190076668595, 3.61904477647992872269217841842, 4.45357866227012854331484553030, 6.77257031568750441009597356176, 7.53498242162851882271322873339, 8.523918692105468880979742786931, 9.272520966926864621056187826752, 10.38863380089069227141539173055, 11.08201541966694667419473648483

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