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Properties

Label 2-336-84.59-c0-0-0
Degree $2$
Conductor $336$
Sign $0.386 - 0.922i$
Analytic cond. $0.167685$
Root an. cond. $0.409494$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + 1.73i·13-s + (−0.5 − 0.866i)19-s − 0.999·21-s + (0.5 − 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−1.49 − 0.866i)39-s − 1.73i·43-s + (−0.499 + 0.866i)49-s + 0.999·57-s + (0.499 − 0.866i)63-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + 1.73i·13-s + (−0.5 − 0.866i)19-s − 0.999·21-s + (0.5 − 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (−1.49 − 0.866i)39-s − 1.73i·43-s + (−0.499 + 0.866i)49-s + 0.999·57-s + (0.499 − 0.866i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(0.167685\)
Root analytic conductor: \(0.409494\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :0),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7028284007\)
\(L(\frac12)\) \(\approx\) \(0.7028284007\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.70936660621357760632839751876, −11.19392642226871800498465004814, −10.12804757903995564835744696774, −9.080799568661909194645623486620, −8.623931761669036879924299159036, −6.96933731186418589627852453076, −6.00634699335077807648720278724, −4.90157129830750161194662772005, −4.07245873583108766166037048020, −2.34444831951433258778789912237, 1.32475020369791494219576842905, 3.14165143099143512860698065346, 4.78279601048968253552291652948, 5.76807003593905703141751472941, 6.87272272618602631039387600295, 7.81163540003019854301079282718, 8.399847951316138107641021008911, 10.11149099981528235191960798606, 10.72516965939401403749378059057, 11.58369911450691363880076425413

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