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Properties

Label 2-336-48.35-c1-0-41
Degree $2$
Conductor $336$
Sign $0.0102 + 0.999i$
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  + (0.153 + 1.40i)2-s + (1.26 − 1.18i)3-s + (−1.95 + 0.432i)4-s + (−2.84 − 2.84i)5-s + (1.85 + 1.59i)6-s − 7-s + (−0.908 − 2.67i)8-s + (0.196 − 2.99i)9-s + (3.55 − 4.43i)10-s + (−3.31 + 3.31i)11-s + (−1.95 + 2.85i)12-s + (−2.31 − 2.31i)13-s + (−0.153 − 1.40i)14-s + (−6.95 − 0.228i)15-s + (3.62 − 1.68i)16-s − 0.814i·17-s + ⋯
L(s)  = 1  + (0.108 + 0.994i)2-s + (0.729 − 0.683i)3-s + (−0.976 + 0.216i)4-s + (−1.27 − 1.27i)5-s + (0.758 + 0.651i)6-s − 0.377·7-s + (−0.321 − 0.947i)8-s + (0.0656 − 0.997i)9-s + (1.12 − 1.40i)10-s + (−1.00 + 1.00i)11-s + (−0.564 + 0.825i)12-s + (−0.641 − 0.641i)13-s + (−0.0410 − 0.375i)14-s + (−1.79 − 0.0590i)15-s + (0.906 − 0.422i)16-s − 0.197i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0102 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0102 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.0102 + 0.999i$
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: \((2,\ 336,\ (\ :1/2),\ 0.0102 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.571449 - 0.565605i\)
\(L(\frac12)\) \(\approx\) \(0.571449 - 0.565605i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.153 - 1.40i)T \)
3 \( 1 + (-1.26 + 1.18i)T \)
7 \( 1 + T \)
good5 \( 1 + (2.84 + 2.84i)T + 5iT^{2} \)
11 \( 1 + (3.31 - 3.31i)T - 11iT^{2} \)
13 \( 1 + (2.31 + 2.31i)T + 13iT^{2} \)
17 \( 1 + 0.814iT - 17T^{2} \)
19 \( 1 + (-4.39 + 4.39i)T - 19iT^{2} \)
23 \( 1 + 3.30iT - 23T^{2} \)
29 \( 1 + (-3.25 + 3.25i)T - 29iT^{2} \)
31 \( 1 - 3.89iT - 31T^{2} \)
37 \( 1 + (3.65 - 3.65i)T - 37iT^{2} \)
41 \( 1 - 3.20T + 41T^{2} \)
43 \( 1 + (-1.63 - 1.63i)T + 43iT^{2} \)
47 \( 1 - 6.70T + 47T^{2} \)
53 \( 1 + (7.69 + 7.69i)T + 53iT^{2} \)
59 \( 1 + (-7.31 + 7.31i)T - 59iT^{2} \)
61 \( 1 + (5.54 + 5.54i)T + 61iT^{2} \)
67 \( 1 + (0.0553 - 0.0553i)T - 67iT^{2} \)
71 \( 1 - 6.50iT - 71T^{2} \)
73 \( 1 - 6.05iT - 73T^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 + (4.25 + 4.25i)T + 83iT^{2} \)
89 \( 1 - 2.31T + 89T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−11.84335909995834654364417194777, −9.948498905215690685939703127538, −9.077091616726216581560236219180, −8.197828507313617610634031172840, −7.62190730658755789725759284413, −6.90581489609235342872010172798, −5.22015290648924028647069339402, −4.46630553518151890132837777286, −3.06623095971429671538944178020, −0.49909355749959647481657195852, 2.65338764019599707950746313150, 3.36329305825428840822740408359, 4.17539091479627337142610075031, 5.60677276368000153651185217693, 7.41383594243874859074557802764, 8.087195829114032432465341268572, 9.202959830434178952575265573002, 10.23568994871791003425312778451, 10.77410303307656236161719798289, 11.56303630486854312952742449879

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