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Properties

Label 2-336-112.109-c1-0-0
Degree $2$
Conductor $336$
Sign $-0.884 + 0.467i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.18 + 0.773i)2-s + (−0.965 + 0.258i)3-s + (0.803 + 1.83i)4-s + (−4.06 − 1.08i)5-s + (−1.34 − 0.440i)6-s + (−2.62 + 0.304i)7-s + (−0.464 + 2.79i)8-s + (0.866 − 0.499i)9-s + (−3.96 − 4.42i)10-s + (−0.970 − 3.62i)11-s + (−1.25 − 1.56i)12-s + (0.0588 + 0.0588i)13-s + (−3.34 − 1.67i)14-s + 4.20·15-s + (−2.70 + 2.94i)16-s + (−2.36 + 4.09i)17-s + ⋯
L(s)  = 1  + (0.837 + 0.546i)2-s + (−0.557 + 0.149i)3-s + (0.401 + 0.915i)4-s + (−1.81 − 0.486i)5-s + (−0.548 − 0.179i)6-s + (−0.993 + 0.114i)7-s + (−0.164 + 0.986i)8-s + (0.288 − 0.166i)9-s + (−1.25 − 1.40i)10-s + (−0.292 − 1.09i)11-s + (−0.360 − 0.450i)12-s + (0.0163 + 0.0163i)13-s + (−0.894 − 0.446i)14-s + 1.08·15-s + (−0.676 + 0.736i)16-s + (−0.573 + 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.884 + 0.467i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.884 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0536711 - 0.216343i\)
\(L(\frac12)\) \(\approx\) \(0.0536711 - 0.216343i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 - 0.773i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (2.62 - 0.304i)T \)
good5 \( 1 + (4.06 + 1.08i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.970 + 3.62i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.0588 - 0.0588i)T + 13iT^{2} \)
17 \( 1 + (2.36 - 4.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.709 + 2.64i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (6.09 - 3.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.33 - 1.33i)T + 29iT^{2} \)
31 \( 1 + (4.34 - 7.53i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.23 - 0.866i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.05iT - 41T^{2} \)
43 \( 1 + (1.31 - 1.31i)T - 43iT^{2} \)
47 \( 1 + (1.56 + 2.71i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.818 + 3.05i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.60 + 5.98i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.0796 - 0.297i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (12.5 - 3.35i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.95iT - 71T^{2} \)
73 \( 1 + (-10.5 - 6.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.19 + 5.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.77 - 1.77i)T + 83iT^{2} \)
89 \( 1 + (-0.469 + 0.270i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.14T + 97T^{2} \)
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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

−12.19304462565268747125320545611, −11.42652463499825690184474809836, −10.67897874716178567478866919193, −8.922997382748775318132127235124, −8.172568024367236673204675325940, −7.18233395039198730688565525682, −6.22033564092781738590156061425, −5.13745661189613985769846086124, −3.96513837822330440207070290218, −3.33322530482999327257603673941, 0.12141811584856841802224284975, 2.66147245645321434015043800560, 3.92695725521372824338580181212, 4.59178054400552907169688657323, 6.13695915739622383346497477791, 7.04420636369128531494040884622, 7.77329978014031214606658771837, 9.555149023093476518091893927583, 10.42075476025785388424038210016, 11.28471106025824185733808137042

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