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Properties

Label 2-336-48.35-c1-0-28
Degree $2$
Conductor $336$
Sign $0.663 + 0.747i$
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  + (0.720 + 1.21i)2-s + (−1.68 − 0.399i)3-s + (−0.962 + 1.75i)4-s + (−2.09 − 2.09i)5-s + (−0.727 − 2.33i)6-s + 7-s + (−2.82 + 0.0901i)8-s + (2.68 + 1.34i)9-s + (1.04 − 4.06i)10-s + (3.61 − 3.61i)11-s + (2.32 − 2.56i)12-s + (−2.99 − 2.99i)13-s + (0.720 + 1.21i)14-s + (2.69 + 4.37i)15-s + (−2.14 − 3.37i)16-s − 2.25i·17-s + ⋯
L(s)  = 1  + (0.509 + 0.860i)2-s + (−0.973 − 0.230i)3-s + (−0.481 + 0.876i)4-s + (−0.938 − 0.938i)5-s + (−0.296 − 0.954i)6-s + 0.377·7-s + (−0.999 + 0.0318i)8-s + (0.893 + 0.449i)9-s + (0.329 − 1.28i)10-s + (1.08 − 1.08i)11-s + (0.670 − 0.741i)12-s + (−0.830 − 0.830i)13-s + (0.192 + 0.325i)14-s + (0.696 + 1.12i)15-s + (−0.536 − 0.844i)16-s − 0.545i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.663 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.663 + 0.747i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.663 + 0.747i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753855 - 0.338856i\)
\(L(\frac12)\) \(\approx\) \(0.753855 - 0.338856i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.720 - 1.21i)T \)
3 \( 1 + (1.68 + 0.399i)T \)
7 \( 1 - T \)
good5 \( 1 + (2.09 + 2.09i)T + 5iT^{2} \)
11 \( 1 + (-3.61 + 3.61i)T - 11iT^{2} \)
13 \( 1 + (2.99 + 2.99i)T + 13iT^{2} \)
17 \( 1 + 2.25iT - 17T^{2} \)
19 \( 1 + (-2.89 + 2.89i)T - 19iT^{2} \)
23 \( 1 + 6.47iT - 23T^{2} \)
29 \( 1 + (7.29 - 7.29i)T - 29iT^{2} \)
31 \( 1 - 6.92iT - 31T^{2} \)
37 \( 1 + (-2.12 + 2.12i)T - 37iT^{2} \)
41 \( 1 + 7.93T + 41T^{2} \)
43 \( 1 + (0.598 + 0.598i)T + 43iT^{2} \)
47 \( 1 - 2.35T + 47T^{2} \)
53 \( 1 + (2.98 + 2.98i)T + 53iT^{2} \)
59 \( 1 + (-1.85 + 1.85i)T - 59iT^{2} \)
61 \( 1 + (-3.40 - 3.40i)T + 61iT^{2} \)
67 \( 1 + (-7.71 + 7.71i)T - 67iT^{2} \)
71 \( 1 + 8.99iT - 71T^{2} \)
73 \( 1 + 9.70iT - 73T^{2} \)
79 \( 1 - 5.37iT - 79T^{2} \)
83 \( 1 + (-4.46 - 4.46i)T + 83iT^{2} \)
89 \( 1 - 4.75T + 89T^{2} \)
97 \( 1 - 3.98T + 97T^{2} \)
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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

−11.79128023299256334044339294137, −10.86500546453111974059596275357, −9.226528876928735027254006011549, −8.384771630613981444659087360088, −7.45119680311891577601389798448, −6.59174725572108808272463872716, −5.27670504996374079131431986165, −4.81371640854498649060856479762, −3.53840888461660183277365718916, −0.58172640287990700819473068626, 1.80878698182778752535706799686, 3.77121116609838277551886651893, 4.30200228957914163124081537895, 5.59897648628358998917215215637, 6.75738828210250645341047117243, 7.60193537349168597757771244413, 9.532246669107809492074005314690, 9.932647581475868982436696800323, 11.19049821713282247078460452323, 11.69647832057550795271410508955

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