Properties

Label 2-336-7.2-c5-0-9
Degree $2$
Conductor $336$
Sign $-0.276 - 0.960i$
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 + (−19.3 + 33.5i)5-s + (87.5 + 95.6i)7-s + (−40.5 + 70.1i)9-s + (−288. − 499. i)11-s + 391.·13-s + 348.·15-s + (664. + 1.15e3i)17-s + (471. − 816. i)19-s + (351. − 1.11e3i)21-s + (−816. + 1.41e3i)23-s + (812. + 1.40e3i)25-s + 729·27-s − 1.46e3·29-s + (−1.95e3 − 3.38e3i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.346 + 0.599i)5-s + (0.674 + 0.737i)7-s + (−0.166 + 0.288i)9-s + (−0.718 − 1.24i)11-s + 0.642·13-s + 0.399·15-s + (0.557 + 0.966i)17-s + (0.299 − 0.518i)19-s + (0.174 − 0.550i)21-s + (−0.321 + 0.557i)23-s + (0.260 + 0.450i)25-s + 0.192·27-s − 0.323·29-s + (−0.365 − 0.633i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.276 - 0.960i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ -0.276 - 0.960i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.087171410\)
\(L(\frac12)\) \(\approx\) \(1.087171410\)
\(L(\frac{7}{2})\) 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 \( 1 + (-87.5 - 95.6i)T \)
good5 \( 1 + (19.3 - 33.5i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (288. + 499. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 391.T + 3.71e5T^{2} \)
17 \( 1 + (-664. - 1.15e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-471. + 816. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (816. - 1.41e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 1.46e3T + 2.05e7T^{2} \)
31 \( 1 + (1.95e3 + 3.38e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-8.15e3 + 1.41e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.31e4T + 1.15e8T^{2} \)
43 \( 1 + 1.47e4T + 1.47e8T^{2} \)
47 \( 1 + (3.40e3 - 5.90e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.00e3 - 1.74e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-2.57e4 - 4.45e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.05e4 - 3.55e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.52e4 - 4.38e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 3.99e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.78e4 - 4.82e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (3.15e4 - 5.46e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 4.55e4T + 3.93e9T^{2} \)
89 \( 1 + (7.84e3 - 1.35e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 3.12e3T + 8.58e9T^{2} \)
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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

−11.27843118485047140817733345075, −10.34339461458288746168943816009, −8.866432142649006486993546397178, −8.124541244781304290015506780566, −7.28891728937252379464928818092, −5.95196967619012783855715792317, −5.44172579933310185463394540904, −3.74468213645877401036924822180, −2.60498203102772553350639142484, −1.22358426558276070032377914069, 0.31561707669727446388179123362, 1.63433161686096948174573854568, 3.40453930163319354171056230492, 4.68045492307606269617540877577, 5.04839945335906316653667957066, 6.62946723123400906210811468591, 7.73848275389500434204338445011, 8.441873915261240413220034636283, 9.784034311281721632784187758470, 10.32091345598006864867236017383

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