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Properties

Label 2-336-21.17-c3-0-29
Degree $2$
Conductor $336$
Sign $0.949 - 0.312i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.18 − 0.403i)3-s + (5.80 + 10.0i)5-s + (18.4 − 2.09i)7-s + (26.6 − 4.17i)9-s + (15.5 + 8.95i)11-s − 62.4i·13-s + (34.1 + 49.7i)15-s + (−10.7 + 18.5i)17-s + (−9.50 + 5.48i)19-s + (94.4 − 18.2i)21-s + (−59.8 + 34.5i)23-s + (−4.82 + 8.35i)25-s + (136. − 32.3i)27-s + 265. i·29-s + (−8.85 − 5.11i)31-s + ⋯
L(s)  = 1  + (0.996 − 0.0775i)3-s + (0.518 + 0.898i)5-s + (0.993 − 0.112i)7-s + (0.987 − 0.154i)9-s + (0.425 + 0.245i)11-s − 1.33i·13-s + (0.587 + 0.855i)15-s + (−0.152 + 0.264i)17-s + (−0.114 + 0.0662i)19-s + (0.981 − 0.189i)21-s + (−0.542 + 0.313i)23-s + (−0.0385 + 0.0668i)25-s + (0.972 − 0.230i)27-s + 1.70i·29-s + (−0.0513 − 0.0296i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.949 - 0.312i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.949 - 0.312i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.362768753\)
\(L(\frac12)\) \(\approx\) \(3.362768753\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-5.18 + 0.403i)T \)
7 \( 1 + (-18.4 + 2.09i)T \)
good5 \( 1 + (-5.80 - 10.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-15.5 - 8.95i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 62.4iT - 2.19e3T^{2} \)
17 \( 1 + (10.7 - 18.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (9.50 - 5.48i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (59.8 - 34.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 265. iT - 2.43e4T^{2} \)
31 \( 1 + (8.85 + 5.11i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (20.8 + 36.0i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 31.0T + 6.89e4T^{2} \)
43 \( 1 - 224.T + 7.95e4T^{2} \)
47 \( 1 + (81.8 + 141. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (456. + 263. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (205. - 356. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-223. + 129. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-161. + 280. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 45.4iT - 3.57e5T^{2} \)
73 \( 1 + (486. + 281. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-144. - 250. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 448.T + 5.71e5T^{2} \)
89 \( 1 + (-280. - 486. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 214. iT - 9.12e5T^{2} \)
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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

−10.83725947355933259561475442369, −10.35026453599062375624644230822, −9.277813286133475782966541458788, −8.256133118036771891612369556132, −7.50956390265439695602286114795, −6.50815196179685020666279626294, −5.16530662550722136492299669603, −3.78605761221020899900156707353, −2.66247382358156692672086102588, −1.50311117982220118156072339866, 1.35216618616488050546472995676, 2.30489252008179561673121799832, 4.09692830462124958078513581955, 4.79971049705225644759869590934, 6.20603339503387610874149613613, 7.51027992279522079213856133620, 8.456985758749109916179006039467, 9.109704077443453514629307394760, 9.803098276314712954507286047096, 11.14520985412450438076639580991

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