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Properties

Label 2-336-336.107-c1-0-59
Degree $2$
Conductor $336$
Sign $-0.704 - 0.709i$
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  + (−0.236 − 1.39i)2-s + (0.301 − 1.70i)3-s + (−1.88 + 0.658i)4-s + (−1.05 − 3.93i)5-s + (−2.44 − 0.0171i)6-s + (−1.10 + 2.40i)7-s + (1.36 + 2.47i)8-s + (−2.81 − 1.02i)9-s + (−5.23 + 2.40i)10-s + (0.104 − 0.388i)11-s + (0.554 + 3.41i)12-s + (−0.933 + 0.933i)13-s + (3.61 + 0.976i)14-s + (−7.03 + 0.613i)15-s + (3.13 − 2.48i)16-s + (3.10 − 1.79i)17-s + ⋯
L(s)  = 1  + (−0.167 − 0.985i)2-s + (0.173 − 0.984i)3-s + (−0.944 + 0.329i)4-s + (−0.471 − 1.76i)5-s + (−0.999 − 0.00699i)6-s + (−0.418 + 0.908i)7-s + (0.482 + 0.875i)8-s + (−0.939 − 0.342i)9-s + (−1.65 + 0.759i)10-s + (0.0313 − 0.117i)11-s + (0.160 + 0.987i)12-s + (−0.258 + 0.258i)13-s + (0.965 + 0.261i)14-s + (−1.81 + 0.158i)15-s + (0.783 − 0.621i)16-s + (0.752 − 0.434i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.704 - 0.709i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.704 - 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.291982 + 0.701849i\)
\(L(\frac12)\) \(\approx\) \(0.291982 + 0.701849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.236 + 1.39i)T \)
3 \( 1 + (-0.301 + 1.70i)T \)
7 \( 1 + (1.10 - 2.40i)T \)
good5 \( 1 + (1.05 + 3.93i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.104 + 0.388i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.933 - 0.933i)T - 13iT^{2} \)
17 \( 1 + (-3.10 + 1.79i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.88 + 1.31i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.72 + 2.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.68 + 3.68i)T + 29iT^{2} \)
31 \( 1 + (-3.47 + 2.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.87 - 1.03i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 + (-6.59 + 6.59i)T - 43iT^{2} \)
47 \( 1 + (1.97 - 3.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.94 - 1.32i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.25 - 4.67i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.83 + 6.83i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.06 + 3.96i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.36iT - 71T^{2} \)
73 \( 1 + (-9.17 + 5.29i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.61 - 1.50i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.95 + 8.95i)T - 83iT^{2} \)
89 \( 1 + (-2.16 + 3.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.74T + 97T^{2} \)
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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

−11.53720057227125354803576379334, −9.753196079624170516424456473465, −9.082370230680527602493490370367, −8.369985678647391858848621096454, −7.58497575887494906325162822225, −5.76807887043246587982431757329, −4.90097552798239893781010368440, −3.42274944927277451144272971071, −1.97285915320457001010982514429, −0.56288161131763091104329832186, 3.30073621580736918270976100330, 3.87127376633521278681266811374, 5.37036631647619960731585874199, 6.50191304372585861860545859533, 7.40872252723798475693966024067, 8.067175937347670969673384337223, 9.599893940237877194238272181393, 10.17997240444756855862464961216, 10.74181550805129332888157645528, 11.92110142627117364965159221518

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