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Properties

Label 2-336-336.125-c1-0-27
Degree $2$
Conductor $336$
Sign $0.948 + 0.317i$
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  + (−1.28 − 0.591i)2-s + (1.44 − 0.947i)3-s + (1.30 + 1.51i)4-s + (−0.639 + 0.639i)5-s + (−2.42 + 0.359i)6-s + (1.10 + 2.40i)7-s + (−0.771 − 2.72i)8-s + (1.20 − 2.74i)9-s + (1.19 − 0.442i)10-s + (2.11 + 2.11i)11-s + (3.32 + 0.971i)12-s + (−0.505 + 0.505i)13-s + (0.000198 − 3.74i)14-s + (−0.320 + 1.53i)15-s + (−0.619 + 3.95i)16-s + 5.88·17-s + ⋯
L(s)  = 1  + (−0.908 − 0.418i)2-s + (0.837 − 0.547i)3-s + (0.650 + 0.759i)4-s + (−0.285 + 0.285i)5-s + (−0.989 + 0.146i)6-s + (0.418 + 0.908i)7-s + (−0.272 − 0.962i)8-s + (0.401 − 0.916i)9-s + (0.379 − 0.140i)10-s + (0.638 + 0.638i)11-s + (0.959 + 0.280i)12-s + (−0.140 + 0.140i)13-s + (5.31e−5 − 0.999i)14-s + (−0.0828 + 0.395i)15-s + (−0.154 + 0.987i)16-s + 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.948 + 0.317i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.948 + 0.317i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20009 - 0.195534i\)
\(L(\frac12)\) \(\approx\) \(1.20009 - 0.195534i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 + 0.591i)T \)
3 \( 1 + (-1.44 + 0.947i)T \)
7 \( 1 + (-1.10 - 2.40i)T \)
good5 \( 1 + (0.639 - 0.639i)T - 5iT^{2} \)
11 \( 1 + (-2.11 - 2.11i)T + 11iT^{2} \)
13 \( 1 + (0.505 - 0.505i)T - 13iT^{2} \)
17 \( 1 - 5.88T + 17T^{2} \)
19 \( 1 + (-1.69 + 1.69i)T - 19iT^{2} \)
23 \( 1 + 2.20T + 23T^{2} \)
29 \( 1 + (2.64 - 2.64i)T - 29iT^{2} \)
31 \( 1 + 8.14iT - 31T^{2} \)
37 \( 1 + (1.18 - 1.18i)T - 37iT^{2} \)
41 \( 1 + 8.70iT - 41T^{2} \)
43 \( 1 + (-2.47 + 2.47i)T - 43iT^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + (-0.335 - 0.335i)T + 53iT^{2} \)
59 \( 1 + (4.05 - 4.05i)T - 59iT^{2} \)
61 \( 1 + (6.80 - 6.80i)T - 61iT^{2} \)
67 \( 1 + (3.80 + 3.80i)T + 67iT^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + 9.79T + 73T^{2} \)
79 \( 1 - 2.78T + 79T^{2} \)
83 \( 1 + (9.64 + 9.64i)T + 83iT^{2} \)
89 \( 1 - 2.04iT - 89T^{2} \)
97 \( 1 - 3.28iT - 97T^{2} \)
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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

−11.85583806712115950035616599773, −10.42518128841560357338298955497, −9.340710417704274529632742622948, −8.910743828047439559415768942135, −7.65203352026596169260901444933, −7.31872318012188069035819952500, −5.89379593739669880838524714384, −3.90669078702804942208049238349, −2.74064780628936591896687894098, −1.57130512845230044401775845175, 1.32889833385019426388556686530, 3.22936191836404843797890528235, 4.50423542657611954957041966179, 5.81465399186431912765070781128, 7.27730868148594449259694743409, 7.969225351362976447211081043703, 8.668091165495333721150440243810, 9.738343562528488367080435831191, 10.34160081867604337729959487811, 11.26807108925831773162402826477

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