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Properties

Label 2-336-7.4-c5-0-3
Degree $2$
Conductor $336$
Sign $-0.266 - 0.963i$
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 + (−43 − 74.4i)5-s + (−24.5 + 127. i)7-s + (−40.5 − 70.1i)9-s + (17 − 29.4i)11-s − 3·13-s + 774.·15-s + (952 − 1.64e3i)17-s + (−744.5 − 1.28e3i)19-s + (−882 − 763. i)21-s + (−112 − 193. i)23-s + (−2.13e3 + 3.69e3i)25-s + 729·27-s − 6.50e3·29-s + (865.5 − 1.49e3i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.769 − 1.33i)5-s + (−0.188 + 0.981i)7-s + (−0.166 − 0.288i)9-s + (0.0423 − 0.0733i)11-s − 0.00492·13-s + 0.888·15-s + (0.798 − 1.38i)17-s + (−0.473 − 0.819i)19-s + (−0.436 − 0.377i)21-s + (−0.0441 − 0.0764i)23-s + (−0.683 + 1.18i)25-s + 0.192·27-s − 1.43·29-s + (0.161 − 0.280i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 336,\ (\ :5/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5907012652\)
\(L(\frac12)\) \(\approx\) \(0.5907012652\)
\(L(\frac{7}{2})\) 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 \( 1 + (24.5 - 127. i)T \)
good5 \( 1 + (43 + 74.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-17 + 29.4i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 3T + 3.71e5T^{2} \)
17 \( 1 + (-952 + 1.64e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (744.5 + 1.28e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (112 + 193. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 6.50e3T + 2.05e7T^{2} \)
31 \( 1 + (-865.5 + 1.49e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-3.81e3 - 6.61e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.54e4T + 1.15e8T^{2} \)
43 \( 1 + 1.84e4T + 1.47e8T^{2} \)
47 \( 1 + (-9.23e3 - 1.59e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-9.97e3 + 1.72e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.59e4 - 2.75e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.88e4 - 4.99e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (3.02e4 - 5.24e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 4.48e4T + 1.80e9T^{2} \)
73 \( 1 + (1.04e4 - 1.80e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (1.52e4 + 2.64e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.10e5T + 3.93e9T^{2} \)
89 \( 1 + (-2.94e4 - 5.10e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.19e5T + 8.58e9T^{2} \)
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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

−11.34259995043504293287818801857, −9.817473108018282603088993141827, −9.112256544030239112704346259981, −8.417022581578570122374377556012, −7.29918099675725587713616994364, −5.79583603946354349110715348309, −5.03876576491856018071939204744, −4.13835196800899946721462848312, −2.74894879406046633554094174919, −0.934503268767898666191670251091, 0.19961444482186620196488331932, 1.77427920326629623305857874680, 3.36347477188803047367950057344, 4.05126388005971583712498279849, 5.82604611142126503147585123740, 6.72080118555812828400635126480, 7.52079214703007463964408120940, 8.145359758083870007747332474261, 9.830461930332676533824696367543, 10.70178923188129224659544950474

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