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Properties

Label 2-336-336.293-c1-0-32
Degree $2$
Conductor $336$
Sign $-0.932 + 0.361i$
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  + (−0.957 − 1.04i)2-s + (−1.52 + 0.830i)3-s + (−0.168 + 1.99i)4-s + (−1.24 − 1.24i)5-s + (2.31 + 0.788i)6-s + (1.10 + 2.40i)7-s + (2.23 − 1.73i)8-s + (1.62 − 2.52i)9-s + (−0.104 + 2.48i)10-s + (0.0609 − 0.0609i)11-s + (−1.39 − 3.16i)12-s + (−1.21 − 1.21i)13-s + (1.44 − 3.45i)14-s + (2.92 + 0.858i)15-s + (−3.94 − 0.670i)16-s − 5.01·17-s + ⋯
L(s)  = 1  + (−0.676 − 0.736i)2-s + (−0.877 + 0.479i)3-s + (−0.0841 + 0.996i)4-s + (−0.555 − 0.555i)5-s + (0.946 + 0.321i)6-s + (0.418 + 0.908i)7-s + (0.790 − 0.612i)8-s + (0.540 − 0.841i)9-s + (−0.0331 + 0.785i)10-s + (0.0183 − 0.0183i)11-s + (−0.403 − 0.914i)12-s + (−0.336 − 0.336i)13-s + (0.385 − 0.922i)14-s + (0.754 + 0.221i)15-s + (−0.985 − 0.167i)16-s − 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.932 + 0.361i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.932 + 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0432784 - 0.231311i\)
\(L(\frac12)\) \(\approx\) \(0.0432784 - 0.231311i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.957 + 1.04i)T \)
3 \( 1 + (1.52 - 0.830i)T \)
7 \( 1 + (-1.10 - 2.40i)T \)
good5 \( 1 + (1.24 + 1.24i)T + 5iT^{2} \)
11 \( 1 + (-0.0609 + 0.0609i)T - 11iT^{2} \)
13 \( 1 + (1.21 + 1.21i)T + 13iT^{2} \)
17 \( 1 + 5.01T + 17T^{2} \)
19 \( 1 + (3.53 + 3.53i)T + 19iT^{2} \)
23 \( 1 - 0.280T + 23T^{2} \)
29 \( 1 + (6.89 + 6.89i)T + 29iT^{2} \)
31 \( 1 + 5.91iT - 31T^{2} \)
37 \( 1 + (-0.0123 - 0.0123i)T + 37iT^{2} \)
41 \( 1 - 3.16iT - 41T^{2} \)
43 \( 1 + (3.09 + 3.09i)T + 43iT^{2} \)
47 \( 1 - 4.07T + 47T^{2} \)
53 \( 1 + (-2.89 + 2.89i)T - 53iT^{2} \)
59 \( 1 + (-7.03 - 7.03i)T + 59iT^{2} \)
61 \( 1 + (9.52 + 9.52i)T + 61iT^{2} \)
67 \( 1 + (5.50 - 5.50i)T - 67iT^{2} \)
71 \( 1 + 8.84T + 71T^{2} \)
73 \( 1 - 7.61T + 73T^{2} \)
79 \( 1 - 1.98T + 79T^{2} \)
83 \( 1 + (-4.67 + 4.67i)T - 83iT^{2} \)
89 \( 1 + 14.9iT - 89T^{2} \)
97 \( 1 - 14.3iT - 97T^{2} \)
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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

−11.36714548431471473774587131955, −10.34479600391372288238157742916, −9.278129149433187025718644684912, −8.657725544665194875622686730499, −7.57000702108020379993768792757, −6.22410880222531081446669889286, −4.84852358894311823566989510446, −4.07224460931858340928517986388, −2.28876572935289893105387304432, −0.22741260596389316862556387815, 1.68208827498836101178581559677, 4.17535621918388910569029088124, 5.25647850626010831019810623464, 6.56522270652130313816861555488, 7.12717509775852524892788979891, 7.85897570875262583391582888299, 9.022261442181228643323317495336, 10.45470308450512625678075758279, 10.81026674561801923165564407640, 11.62057514570005590335900226488

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