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Properties

Label 2-336-7.4-c5-0-8
Degree $2$
Conductor $336$
Sign $-0.394 - 0.919i$
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 + (52.0 + 90.2i)5-s + (7.12 − 129. i)7-s + (−40.5 − 70.1i)9-s + (−248. + 430. i)11-s − 206.·13-s + 937.·15-s + (−31.5 + 54.6i)17-s + (661. + 1.14e3i)19-s + (−976. − 638. i)21-s + (−97.2 − 168. i)23-s + (−3.86e3 + 6.69e3i)25-s − 729·27-s + 4.32e3·29-s + (−3.76e3 + 6.51e3i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.931 + 1.61i)5-s + (0.0549 − 0.998i)7-s + (−0.166 − 0.288i)9-s + (−0.620 + 1.07i)11-s − 0.338·13-s + 1.07·15-s + (−0.0265 + 0.0459i)17-s + (0.420 + 0.728i)19-s + (−0.483 − 0.315i)21-s + (−0.0383 − 0.0663i)23-s + (−1.23 + 2.14i)25-s − 0.192·27-s + 0.954·29-s + (−0.703 + 1.21i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.919i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.394 - 0.919i$
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: \((2,\ 336,\ (\ :5/2),\ -0.394 - 0.919i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.763636343\)
\(L(\frac12)\) \(\approx\) \(1.763636343\)
\(L(\frac{7}{2})\) 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 \( 1 + (-7.12 + 129. i)T \)
good5 \( 1 + (-52.0 - 90.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (248. - 430. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 206.T + 3.71e5T^{2} \)
17 \( 1 + (31.5 - 54.6i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-661. - 1.14e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (97.2 + 168. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 4.32e3T + 2.05e7T^{2} \)
31 \( 1 + (3.76e3 - 6.51e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (5.17e3 + 8.96e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 4.18e3T + 1.15e8T^{2} \)
43 \( 1 + 5.96e3T + 1.47e8T^{2} \)
47 \( 1 + (2.19e3 + 3.80e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (8.89e3 - 1.54e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.75e3 + 3.03e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-5.31e3 - 9.20e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (6.63e3 - 1.14e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.88e4T + 1.80e9T^{2} \)
73 \( 1 + (1.56e4 - 2.71e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.97e4 - 3.42e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.02e5T + 3.93e9T^{2} \)
89 \( 1 + (-5.64e4 - 9.77e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 3.03e4T + 8.58e9T^{2} \)
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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

−10.55549403351234561911282230569, −10.38001186999911711432456188731, −9.400089696856744440470346369169, −7.81249782924147399202902877818, −7.12846502799680877494371002520, −6.52279061615961695084111765214, −5.22232181513587029737065669142, −3.62753007918037661067905028404, −2.56178306353535897898732439532, −1.57395270163067644911969534405, 0.40085532481543234424406578077, 1.85303699278435654611264686534, 3.01690650015013815391708820836, 4.73208489653738423238167792120, 5.31276800734621656614162299208, 6.15672240609739575303439843213, 8.044299338222740249873105824945, 8.721524192441726061662076077048, 9.340277495006368861751702116988, 10.15021893037594569110077210981

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