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Properties

Label 2-336-112.37-c1-0-0
Degree $2$
Conductor $336$
Sign $0.556 - 0.830i$
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  + (−0.485 − 1.32i)2-s + (−0.965 − 0.258i)3-s + (−1.52 + 1.28i)4-s + (−0.879 + 0.235i)5-s + (0.125 + 1.40i)6-s + (−1.69 − 2.03i)7-s + (2.45 + 1.40i)8-s + (0.866 + 0.499i)9-s + (0.740 + 1.05i)10-s + (0.0635 − 0.237i)11-s + (1.81 − 0.850i)12-s + (−4.10 + 4.10i)13-s + (−1.88 + 3.23i)14-s + 0.911·15-s + (0.672 − 3.94i)16-s + (3.38 + 5.86i)17-s + ⋯
L(s)  = 1  + (−0.343 − 0.939i)2-s + (−0.557 − 0.149i)3-s + (−0.764 + 0.644i)4-s + (−0.393 + 0.105i)5-s + (0.0511 + 0.575i)6-s + (−0.639 − 0.769i)7-s + (0.868 + 0.496i)8-s + (0.288 + 0.166i)9-s + (0.234 + 0.333i)10-s + (0.0191 − 0.0715i)11-s + (0.522 − 0.245i)12-s + (−1.13 + 1.13i)13-s + (−0.502 + 0.864i)14-s + 0.235·15-s + (0.168 − 0.985i)16-s + (0.821 + 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.556 - 0.830i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.556 - 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337470 + 0.180102i\)
\(L(\frac12)\) \(\approx\) \(0.337470 + 0.180102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.485 + 1.32i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 + (1.69 + 2.03i)T \)
good5 \( 1 + (0.879 - 0.235i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.0635 + 0.237i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (4.10 - 4.10i)T - 13iT^{2} \)
17 \( 1 + (-3.38 - 5.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.39 - 5.20i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.95 - 2.86i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0308 + 0.0308i)T - 29iT^{2} \)
31 \( 1 + (4.53 + 7.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.30 - 1.15i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.70iT - 41T^{2} \)
43 \( 1 + (1.74 + 1.74i)T + 43iT^{2} \)
47 \( 1 + (3.98 - 6.90i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.03 - 3.87i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.17 - 4.39i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.52 - 9.43i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (4.40 + 1.18i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 15.2iT - 71T^{2} \)
73 \( 1 + (4.56 - 2.63i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.25 + 7.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.0 - 11.0i)T - 83iT^{2} \)
89 \( 1 + (3.86 + 2.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.40T + 97T^{2} \)
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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

−11.71878616457395275391175940804, −10.77105408807354729115973540234, −10.00417545520872317800911649221, −9.270130921228092023747936697364, −7.81377326620342835144824615395, −7.20070893570566436809616416254, −5.74667305996147128622559027267, −4.29322958947443087057069301980, −3.46700041898499395443460379826, −1.61791188425844926099017830906, 0.33101033419366802801457896499, 3.07401138349634279019649260233, 4.97722834205209068629464539609, 5.33392183046765898510669100860, 6.73376830694350994120996738501, 7.40004236145069502153831159753, 8.579026102012923446074808721385, 9.542722092396862678940581856471, 10.15816898037838997001098106413, 11.41171677295445289516646507844

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