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Properties

Label 2-336-21.5-c3-0-32
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: $\chi_{336} (257, \cdot )$
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

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

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