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Properties

Label 2-336-48.35-c1-0-6
Degree $2$
Conductor $336$
Sign $0.454 - 0.890i$
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  + (−1.25 + 0.648i)2-s + (−0.629 − 1.61i)3-s + (1.15 − 1.62i)4-s + (3.08 + 3.08i)5-s + (1.83 + 1.61i)6-s − 7-s + (−0.400 + 2.79i)8-s + (−2.20 + 2.03i)9-s + (−5.88 − 1.87i)10-s + (−1.97 + 1.97i)11-s + (−3.35 − 0.844i)12-s + (3.27 + 3.27i)13-s + (1.25 − 0.648i)14-s + (3.03 − 6.92i)15-s + (−1.31 − 3.77i)16-s − 2.57i·17-s + ⋯
L(s)  = 1  + (−0.888 + 0.458i)2-s + (−0.363 − 0.931i)3-s + (0.579 − 0.814i)4-s + (1.38 + 1.38i)5-s + (0.750 + 0.661i)6-s − 0.377·7-s + (−0.141 + 0.989i)8-s + (−0.735 + 0.677i)9-s + (−1.85 − 0.593i)10-s + (−0.596 + 0.596i)11-s + (−0.969 − 0.243i)12-s + (0.908 + 0.908i)13-s + (0.335 − 0.173i)14-s + (0.784 − 1.78i)15-s + (−0.328 − 0.944i)16-s − 0.623i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.454 - 0.890i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.454 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.752369 + 0.460685i\)
\(L(\frac12)\) \(\approx\) \(0.752369 + 0.460685i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 - 0.648i)T \)
3 \( 1 + (0.629 + 1.61i)T \)
7 \( 1 + T \)
good5 \( 1 + (-3.08 - 3.08i)T + 5iT^{2} \)
11 \( 1 + (1.97 - 1.97i)T - 11iT^{2} \)
13 \( 1 + (-3.27 - 3.27i)T + 13iT^{2} \)
17 \( 1 + 2.57iT - 17T^{2} \)
19 \( 1 + (1.18 - 1.18i)T - 19iT^{2} \)
23 \( 1 - 1.21iT - 23T^{2} \)
29 \( 1 + (-1.33 + 1.33i)T - 29iT^{2} \)
31 \( 1 - 5.84iT - 31T^{2} \)
37 \( 1 + (-5.18 + 5.18i)T - 37iT^{2} \)
41 \( 1 - 0.688T + 41T^{2} \)
43 \( 1 + (-0.649 - 0.649i)T + 43iT^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + (3.84 + 3.84i)T + 53iT^{2} \)
59 \( 1 + (-1.99 + 1.99i)T - 59iT^{2} \)
61 \( 1 + (6.62 + 6.62i)T + 61iT^{2} \)
67 \( 1 + (-4.50 + 4.50i)T - 67iT^{2} \)
71 \( 1 - 3.91iT - 71T^{2} \)
73 \( 1 + 9.83iT - 73T^{2} \)
79 \( 1 - 3.86iT - 79T^{2} \)
83 \( 1 + (-2.99 - 2.99i)T + 83iT^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−11.30110776760921122537307331136, −10.73675807341784568656823496296, −9.865512406961956955894892016428, −9.011029055522062002916140971789, −7.63172913230772281124772173316, −6.82368087741321393099135581306, −6.29036851168626625687649270790, −5.43277554097980254068855899980, −2.70462018684108869912783037358, −1.75231105880770624261047266276, 0.895753451142560204340948782843, 2.72613198614631753606629216544, 4.23351408458431575263617849120, 5.63084876032663229467651363400, 6.18920144035361655704924247752, 8.230397513492637302800435122309, 8.798230745481739802750155176404, 9.613590101059480944350524466467, 10.32792132736549540861952752289, 10.98271454885951585156166515874

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