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Properties

Label 2-336-48.11-c1-0-32
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  + (1.38 + 0.302i)2-s + (0.985 + 1.42i)3-s + (1.81 + 0.836i)4-s + (0.766 − 0.766i)5-s + (0.929 + 2.26i)6-s − 7-s + (2.25 + 1.70i)8-s + (−1.05 + 2.80i)9-s + (1.29 − 0.826i)10-s + (−3.09 − 3.09i)11-s + (0.598 + 3.41i)12-s + (0.262 − 0.262i)13-s + (−1.38 − 0.302i)14-s + (1.84 + 0.336i)15-s + (2.60 + 3.03i)16-s − 2.68i·17-s + ⋯
L(s)  = 1  + (0.976 + 0.213i)2-s + (0.568 + 0.822i)3-s + (0.908 + 0.418i)4-s + (0.342 − 0.342i)5-s + (0.379 + 0.925i)6-s − 0.377·7-s + (0.797 + 0.602i)8-s + (−0.353 + 0.935i)9-s + (0.408 − 0.261i)10-s + (−0.932 − 0.932i)11-s + (0.172 + 0.984i)12-s + (0.0728 − 0.0728i)13-s + (−0.369 − 0.0808i)14-s + (0.476 + 0.0869i)15-s + (0.650 + 0.759i)16-s − 0.651i·17-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: $\chi_{336} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.41918 + 1.19927i\)
\(L(\frac12)\) \(\approx\) \(2.41918 + 1.19927i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.302i)T \)
3 \( 1 + (-0.985 - 1.42i)T \)
7 \( 1 + T \)
good5 \( 1 + (-0.766 + 0.766i)T - 5iT^{2} \)
11 \( 1 + (3.09 + 3.09i)T + 11iT^{2} \)
13 \( 1 + (-0.262 + 0.262i)T - 13iT^{2} \)
17 \( 1 + 2.68iT - 17T^{2} \)
19 \( 1 + (2.44 + 2.44i)T + 19iT^{2} \)
23 \( 1 + 3.24iT - 23T^{2} \)
29 \( 1 + (-3.28 - 3.28i)T + 29iT^{2} \)
31 \( 1 + 3.76iT - 31T^{2} \)
37 \( 1 + (-4.88 - 4.88i)T + 37iT^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + (0.938 - 0.938i)T - 43iT^{2} \)
47 \( 1 + 0.764T + 47T^{2} \)
53 \( 1 + (-4.14 + 4.14i)T - 53iT^{2} \)
59 \( 1 + (-2.63 - 2.63i)T + 59iT^{2} \)
61 \( 1 + (-10.7 + 10.7i)T - 61iT^{2} \)
67 \( 1 + (8.41 + 8.41i)T + 67iT^{2} \)
71 \( 1 - 13.7iT - 71T^{2} \)
73 \( 1 + 6.79iT - 73T^{2} \)
79 \( 1 - 3.83iT - 79T^{2} \)
83 \( 1 + (-0.107 + 0.107i)T - 83iT^{2} \)
89 \( 1 + 9.00T + 89T^{2} \)
97 \( 1 - 18.1T + 97T^{2} \)
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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

−11.65661833876117769632961785626, −10.81573272690027053229272742074, −9.974227140959940703988419247239, −8.765896310821336505392025059932, −8.000693605729208591283102058714, −6.67504589594596678494997558954, −5.46634987014405048762903007337, −4.76365832290335884648964036593, −3.43919859823469685243033764777, −2.52612670193980706094584161981, 1.88468609873415800063172056417, 2.85450646815632225457192460354, 4.13112691674785524670994043015, 5.62107739806546695379471787487, 6.53272164221095703426549440113, 7.34754902786846101968707134249, 8.380135592314013615578857459506, 9.879187964313341351393414995119, 10.47054786832164769233687863983, 11.80444308265158393907293844706

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