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Properties

Label 2-336-7.4-c5-0-32
Degree $2$
Conductor $336$
Sign $-0.770 + 0.636i$
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 + (9.56 + 16.5i)5-s + (126. − 29.5i)7-s + (−40.5 − 70.1i)9-s + (−316. + 547. i)11-s − 626.·13-s + 172.·15-s + (461. − 800. i)17-s + (−1.20e3 − 2.08e3i)19-s + (337. − 1.11e3i)21-s + (−192. − 332. i)23-s + (1.37e3 − 2.38e3i)25-s − 729·27-s − 2.70e3·29-s + (−615. + 1.06e3i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.171 + 0.296i)5-s + (0.973 − 0.228i)7-s + (−0.166 − 0.288i)9-s + (−0.787 + 1.36i)11-s − 1.02·13-s + 0.197·15-s + (0.387 − 0.671i)17-s + (−0.763 − 1.32i)19-s + (0.166 − 0.552i)21-s + (−0.0757 − 0.131i)23-s + (0.441 − 0.764i)25-s − 0.192·27-s − 0.598·29-s + (−0.115 + 0.199i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.636i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.770 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.770 + 0.636i$
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: \((2,\ 336,\ (\ :5/2),\ -0.770 + 0.636i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.184911381\)
\(L(\frac12)\) \(\approx\) \(1.184911381\)
\(L(\frac{7}{2})\) 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 \( 1 + (-126. + 29.5i)T \)
good5 \( 1 + (-9.56 - 16.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (316. - 547. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 626.T + 3.71e5T^{2} \)
17 \( 1 + (-461. + 800. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.20e3 + 2.08e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (192. + 332. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 2.70e3T + 2.05e7T^{2} \)
31 \( 1 + (615. - 1.06e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (1.69e3 + 2.92e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.90e4T + 1.15e8T^{2} \)
43 \( 1 + 2.15e4T + 1.47e8T^{2} \)
47 \( 1 + (4.71e3 + 8.17e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.14e4 + 1.98e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-5.57e3 + 9.65e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.67e4 + 2.89e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.49e3 - 2.59e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 7.49e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.28e4 + 2.22e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-3.80e4 - 6.59e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 4.05e3T + 3.93e9T^{2} \)
89 \( 1 + (5.60e4 + 9.70e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.00e5T + 8.58e9T^{2} \)
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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

−10.34444898771063311402536754799, −9.483159806634226921841663433105, −8.305965912981874553566289403820, −7.39376517358370508703987409136, −6.85779911346626016742214335430, −5.20955420649083760209807077479, −4.49136767420288490646576628846, −2.66578222343428417326067598070, −1.93168244879969479736777951544, −0.27289119251384512250982275881, 1.47889002129844324626029704308, 2.78137522821876232066809322349, 4.05904187164808797660312251080, 5.22751676748694207881700762075, 5.89282281922891840252656108088, 7.65596424034291015362215667680, 8.268178885300045380744375832105, 9.125299313622474446948837571115, 10.27974779271902120515194105069, 10.91109070037534635881408732401

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