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Properties

Label 2-336-336.173-c1-0-38
Degree $2$
Conductor $336$
Sign $0.806 - 0.591i$
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  + (1.16 + 0.798i)2-s + (1.06 − 1.36i)3-s + (0.725 + 1.86i)4-s + (0.729 + 2.72i)5-s + (2.33 − 0.741i)6-s + (0.617 − 2.57i)7-s + (−0.641 + 2.75i)8-s + (−0.725 − 2.91i)9-s + (−1.32 + 3.75i)10-s + (−0.126 + 0.473i)11-s + (3.31 + 0.997i)12-s + (−1.61 + 1.61i)13-s + (2.77 − 2.50i)14-s + (4.49 + 1.90i)15-s + (−2.94 + 2.70i)16-s + (0.919 + 1.59i)17-s + ⋯
L(s)  = 1  + (0.825 + 0.564i)2-s + (0.615 − 0.787i)3-s + (0.362 + 0.931i)4-s + (0.326 + 1.21i)5-s + (0.953 − 0.302i)6-s + (0.233 − 0.972i)7-s + (−0.226 + 0.973i)8-s + (−0.241 − 0.970i)9-s + (−0.417 + 1.18i)10-s + (−0.0382 + 0.142i)11-s + (0.957 + 0.288i)12-s + (−0.448 + 0.448i)13-s + (0.741 − 0.670i)14-s + (1.15 + 0.492i)15-s + (−0.737 + 0.675i)16-s + (0.222 + 0.386i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.806 - 0.591i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.806 - 0.591i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39871 + 0.785144i\)
\(L(\frac12)\) \(\approx\) \(2.39871 + 0.785144i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 0.798i)T \)
3 \( 1 + (-1.06 + 1.36i)T \)
7 \( 1 + (-0.617 + 2.57i)T \)
good5 \( 1 + (-0.729 - 2.72i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.126 - 0.473i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.61 - 1.61i)T - 13iT^{2} \)
17 \( 1 + (-0.919 - 1.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.02 + 7.54i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.49 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.89 - 2.89i)T - 29iT^{2} \)
31 \( 1 + (-9.09 + 5.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.460 + 1.71i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.10iT - 41T^{2} \)
43 \( 1 + (6.34 - 6.34i)T - 43iT^{2} \)
47 \( 1 + (0.607 - 1.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.913 + 3.41i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.84 + 1.02i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.66 - 6.19i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.532 - 1.98i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.11T + 71T^{2} \)
73 \( 1 + (2.86 + 4.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.97 - 8.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.94 - 9.94i)T + 83iT^{2} \)
89 \( 1 + (12.6 + 7.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.90iT - 97T^{2} \)
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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

−11.72890309896363631102072811570, −10.99565170012844298052371716296, −9.789176766757360569744783348975, −8.471323408407888711955295303497, −7.37814717632626677885252337572, −6.96601998913065101982090375530, −6.13738595269215241506308515788, −4.51112413091963408508318536920, −3.29275615711127453678858490858, −2.24700925848346644638702722491, 1.84503707496356719218631402265, 3.11281806077173220940345756845, 4.48104042151006445653802591843, 5.19209791822000170100695007825, 6.03664357433413183897421290615, 8.048014367670822244608703400400, 8.801130451849341274978302713346, 9.790382607881749099815856696697, 10.36496703734044184667179791442, 11.74477591732251408925216035838

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