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Properties

Label 2-336-7.4-c5-0-35
Degree $2$
Conductor $336$
Sign $-0.968 + 0.250i$
Analytic cond. $53.8889$
Root an. cond. $7.34091$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.5 − 7.79i)3-s + (−11.0 − 19.1i)5-s + (−126. − 26.6i)7-s + (−40.5 − 70.1i)9-s + (208. − 360. i)11-s + 797.·13-s − 198.·15-s + (687. − 1.19e3i)17-s + (1.15e3 + 2.00e3i)19-s + (−778. + 869. i)21-s + (−477. − 827. i)23-s + (1.31e3 − 2.28e3i)25-s − 729·27-s − 7.03e3·29-s + (630. − 1.09e3i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.197 − 0.341i)5-s + (−0.978 − 0.205i)7-s + (−0.166 − 0.288i)9-s + (0.519 − 0.899i)11-s + 1.30·13-s − 0.227·15-s + (0.577 − 0.999i)17-s + (0.734 + 1.27i)19-s + (−0.385 + 0.430i)21-s + (−0.188 − 0.326i)23-s + (0.422 − 0.731i)25-s − 0.192·27-s − 1.55·29-s + (0.117 − 0.204i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.968 + 0.250i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.316998916\)
\(L(\frac12)\) \(\approx\) \(1.316998916\)
\(L(\frac{7}{2})\) 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 \( 1 + (-4.5 + 7.79i)T \)
7 \( 1 + (126. + 26.6i)T \)
good5 \( 1 + (11.0 + 19.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-208. + 360. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 797.T + 3.71e5T^{2} \)
17 \( 1 + (-687. + 1.19e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.15e3 - 2.00e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (477. + 827. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 7.03e3T + 2.05e7T^{2} \)
31 \( 1 + (-630. + 1.09e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (4.88e3 + 8.46e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 5.40e3T + 1.15e8T^{2} \)
43 \( 1 + 1.96e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.02e3 - 1.78e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (9.01e3 - 1.56e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-3.71e3 + 6.43e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.74e3 + 3.02e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-7.92e3 + 1.37e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 5.81e4T + 1.80e9T^{2} \)
73 \( 1 + (1.95e4 - 3.38e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-4.88e3 - 8.45e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 7.03e4T + 3.93e9T^{2} \)
89 \( 1 + (7.21e4 + 1.24e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 7.93e4T + 8.58e9T^{2} \)
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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

−10.21634098439687661354715587828, −9.226383918285151176647271872339, −8.455958252020127116657901535127, −7.46045476203689725062209569663, −6.37797623537009223601277914757, −5.60952194428769375511132234178, −3.83235626691317496463558393072, −3.16981203884020678710840393660, −1.40741748889352541167801562551, −0.34035693491137297772179044406, 1.54572592599928544773981665960, 3.18652749002496854757248991718, 3.79375999447260143362188845650, 5.22573297389418479673743106928, 6.39871181117403138415471448182, 7.25176614841748258323871493723, 8.530678442921327649579601560416, 9.362013074810042908633863679676, 10.10197503684529526236258207510, 11.08292648495588681860366588438

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