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Properties

Label 2-336-48.11-c1-0-42
Degree $2$
Conductor $336$
Sign $-0.962 + 0.269i$
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  + (0.806 − 1.16i)2-s + (−1.73 − 0.0806i)3-s + (−0.697 − 1.87i)4-s + (1.66 − 1.66i)5-s + (−1.48 + 1.94i)6-s − 7-s + (−2.73 − 0.702i)8-s + (2.98 + 0.279i)9-s + (−0.588 − 3.27i)10-s + (−3.32 − 3.32i)11-s + (1.05 + 3.29i)12-s + (−0.938 + 0.938i)13-s + (−0.806 + 1.16i)14-s + (−3.00 + 2.74i)15-s + (−3.02 + 2.61i)16-s − 0.811i·17-s + ⋯
L(s)  = 1  + (0.570 − 0.821i)2-s + (−0.998 − 0.0465i)3-s + (−0.348 − 0.937i)4-s + (0.743 − 0.743i)5-s + (−0.608 + 0.793i)6-s − 0.377·7-s + (−0.968 − 0.248i)8-s + (0.995 + 0.0930i)9-s + (−0.186 − 1.03i)10-s + (−1.00 − 1.00i)11-s + (0.304 + 0.952i)12-s + (−0.260 + 0.260i)13-s + (−0.215 + 0.310i)14-s + (−0.777 + 0.707i)15-s + (−0.756 + 0.653i)16-s − 0.196i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.962 + 0.269i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.962 + 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147048 - 1.06973i\)
\(L(\frac12)\) \(\approx\) \(0.147048 - 1.06973i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.806 + 1.16i)T \)
3 \( 1 + (1.73 + 0.0806i)T \)
7 \( 1 + T \)
good5 \( 1 + (-1.66 + 1.66i)T - 5iT^{2} \)
11 \( 1 + (3.32 + 3.32i)T + 11iT^{2} \)
13 \( 1 + (0.938 - 0.938i)T - 13iT^{2} \)
17 \( 1 + 0.811iT - 17T^{2} \)
19 \( 1 + (-0.974 - 0.974i)T + 19iT^{2} \)
23 \( 1 + 7.19iT - 23T^{2} \)
29 \( 1 + (2.21 + 2.21i)T + 29iT^{2} \)
31 \( 1 - 6.74iT - 31T^{2} \)
37 \( 1 + (3.62 + 3.62i)T + 37iT^{2} \)
41 \( 1 - 6.63T + 41T^{2} \)
43 \( 1 + (-8.58 + 8.58i)T - 43iT^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + (-5.01 + 5.01i)T - 53iT^{2} \)
59 \( 1 + (-3.36 - 3.36i)T + 59iT^{2} \)
61 \( 1 + (-9.07 + 9.07i)T - 61iT^{2} \)
67 \( 1 + (-2.30 - 2.30i)T + 67iT^{2} \)
71 \( 1 + 1.63iT - 71T^{2} \)
73 \( 1 + 1.89iT - 73T^{2} \)
79 \( 1 - 13.3iT - 79T^{2} \)
83 \( 1 + (5.35 - 5.35i)T - 83iT^{2} \)
89 \( 1 + 7.32T + 89T^{2} \)
97 \( 1 + 19.2T + 97T^{2} \)
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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

−11.05434782603618634098679009826, −10.49474748794368491038172775484, −9.602158212524394647083489117603, −8.635542621714280178499144379747, −6.91084870595808224788262488359, −5.66830512573300333157145043145, −5.35665741867373292877493755613, −4.08011401442739250540131519748, −2.39529069075002448989435907844, −0.70625213244856380658874530088, 2.58235209167113638147570543721, 4.18468716135200015391883171419, 5.41151393981431451966149995266, 5.96402262051621830457667782388, 7.09842486018811374589554979998, 7.61642890663973307458071865203, 9.409341386621850022988013496634, 10.10209980222225539460357208964, 11.09995710138420102393507113003, 12.17666150828563260143086985499

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