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Properties

Label 2-336-7.4-c1-0-5
Degree $2$
Conductor $336$
Sign $0.605 + 0.795i$
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  + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 2.59i)7-s + (−0.499 − 0.866i)9-s + (1.5 − 2.59i)11-s + 4·13-s + 0.999·15-s + (−2 − 3.46i)19-s + (−2.5 − 0.866i)21-s + (4 + 6.92i)23-s + (2 − 3.46i)25-s − 0.999·27-s − 3·29-s + (−2.5 + 4.33i)31-s + (−1.5 − 2.59i)33-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.223 + 0.387i)5-s + (−0.188 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (0.452 − 0.783i)11-s + 1.10·13-s + 0.258·15-s + (−0.458 − 0.794i)19-s + (−0.545 − 0.188i)21-s + (0.834 + 1.44i)23-s + (0.400 − 0.692i)25-s − 0.192·27-s − 0.557·29-s + (−0.449 + 0.777i)31-s + (−0.261 − 0.452i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35293 - 0.670688i\)
\(L(\frac12)\) \(\approx\) \(1.35293 - 0.670688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (-5 - 8.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.5 - 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3 + 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7T + 83T^{2} \)
89 \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−11.04435583558840974334112606251, −10.87646019263452338569845838805, −9.416301749636659985187879958376, −8.654119078104937237731057740227, −7.46330659107816661056272453218, −6.70819660452786775820537604428, −5.74372940352540178256403532821, −4.04881298692674209535868395267, −3.04168786639307196021575513186, −1.20686075024325120967931599859, 1.92782916575196486636189341960, 3.43472352107102871488601814606, 4.66742853115751436336700555480, 5.73921741878505684235593132581, 6.75880415736497526996388806916, 8.275549315163550816236887973917, 8.913724464164031676128686744716, 9.683454707047541994380299294704, 10.70482832250080415851759366785, 11.68679580528501483916401391011

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