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Properties

Label 2-336-84.11-c3-0-30
Degree $2$
Conductor $336$
Sign $0.811 + 0.584i$
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  + (−5.19 − 0.158i)3-s + (17.5 − 10.1i)5-s + (18.5 + 0.135i)7-s + (26.9 + 1.64i)9-s + (1.55 − 2.69i)11-s + 32.4·13-s + (−92.7 + 49.8i)15-s + (14.0 + 8.09i)17-s + (−66.3 + 38.3i)19-s + (−96.1 − 3.63i)21-s + (16.6 + 28.8i)23-s + (142. − 247. i)25-s + (−139. − 12.7i)27-s + 173. i·29-s + (164. + 94.7i)31-s + ⋯
L(s)  = 1  + (−0.999 − 0.0304i)3-s + (1.56 − 0.906i)5-s + (0.999 + 0.00733i)7-s + (0.998 + 0.0608i)9-s + (0.0426 − 0.0738i)11-s + 0.691·13-s + (−1.59 + 0.858i)15-s + (0.200 + 0.115i)17-s + (−0.801 + 0.462i)19-s + (−0.999 − 0.0377i)21-s + (0.151 + 0.261i)23-s + (1.14 − 1.97i)25-s + (−0.995 − 0.0911i)27-s + 1.11i·29-s + (0.950 + 0.548i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.811 + 0.584i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.811 + 0.584i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.127318401\)
\(L(\frac12)\) \(\approx\) \(2.127318401\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.19 + 0.158i)T \)
7 \( 1 + (-18.5 - 0.135i)T \)
good5 \( 1 + (-17.5 + 10.1i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-1.55 + 2.69i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 32.4T + 2.19e3T^{2} \)
17 \( 1 + (-14.0 - 8.09i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (66.3 - 38.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-16.6 - 28.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 173. iT - 2.43e4T^{2} \)
31 \( 1 + (-164. - 94.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (129. + 224. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 149. iT - 6.89e4T^{2} \)
43 \( 1 + 495. iT - 7.95e4T^{2} \)
47 \( 1 + (-236. - 409. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-482. - 278. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-262. + 454. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (111. + 192. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-241. - 139. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.06e3T + 3.57e5T^{2} \)
73 \( 1 + (-144. + 250. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-641. + 370. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 628.T + 5.71e5T^{2} \)
89 \( 1 + (1.13e3 - 657. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.22e3T + 9.12e5T^{2} \)
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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

−10.80505139357119544784077512565, −10.37057089681902818161286461718, −9.170819439950713170338230783632, −8.410465165348009311703888763596, −6.94477048751635732041316463959, −5.80154207223010479221888366567, −5.34473702413096732548587621119, −4.27538161207573440148620987481, −1.92395892431196434542395337375, −1.07362891801850758664592858976, 1.28050296864350321748560040995, 2.47076071052899500160759831086, 4.38013756719944833880777758644, 5.50497110709122645252268310765, 6.24619894619930272718071016310, 7.03317195640768728939707153084, 8.423404391040342013887958108083, 9.736304107725319576047541749874, 10.37182175317741245925459149566, 11.12190610462412628093042573468

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