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Properties

Label 2-336-21.5-c1-0-9
Degree $2$
Conductor $336$
Sign $0.834 + 0.551i$
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  + (−0.642 + 1.60i)3-s + (1.28 − 2.23i)5-s + (0.203 − 2.63i)7-s + (−2.17 − 2.06i)9-s + (1.43 − 0.826i)11-s − 5.71i·13-s + (2.76 + 3.50i)15-s + (3.79 + 6.56i)17-s + (−2.58 − 1.49i)19-s + (4.11 + 2.02i)21-s + (0.249 + 0.143i)23-s + (−0.825 − 1.43i)25-s + (4.72 − 2.16i)27-s + 2.05i·29-s + (5.21 − 3.00i)31-s + ⋯
L(s)  = 1  + (−0.371 + 0.928i)3-s + (0.576 − 0.998i)5-s + (0.0768 − 0.997i)7-s + (−0.724 − 0.689i)9-s + (0.431 − 0.249i)11-s − 1.58i·13-s + (0.713 + 0.906i)15-s + (0.919 + 1.59i)17-s + (−0.594 − 0.343i)19-s + (0.897 + 0.441i)21-s + (0.0519 + 0.0300i)23-s + (−0.165 − 0.286i)25-s + (0.908 − 0.417i)27-s + 0.382i·29-s + (0.936 − 0.540i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.834 + 0.551i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.834 + 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20566 - 0.362307i\)
\(L(\frac12)\) \(\approx\) \(1.20566 - 0.362307i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.642 - 1.60i)T \)
7 \( 1 + (-0.203 + 2.63i)T \)
good5 \( 1 + (-1.28 + 2.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.43 + 0.826i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.71iT - 13T^{2} \)
17 \( 1 + (-3.79 - 6.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.58 + 1.49i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.249 - 0.143i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.05iT - 29T^{2} \)
31 \( 1 + (-5.21 + 3.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.877 - 1.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.28T + 41T^{2} \)
43 \( 1 + 2.46T + 43T^{2} \)
47 \( 1 + (-0.186 + 0.323i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.73 - 3.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.89 + 8.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.889 - 0.513i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.18 - 2.04i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 + (3.30 - 1.90i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.56 - 7.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.65T + 83T^{2} \)
89 \( 1 + (-7.25 + 12.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.43iT - 97T^{2} \)
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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

−11.22811227788553747382750907880, −10.38000141971606750985034350488, −9.871599355052059887123990961276, −8.718999703460147405953944950956, −7.947451140087494299677852387073, −6.28197980877308278748843154787, −5.46128568045865471165382811365, −4.45292486738036702010474447560, −3.39679287658090198475772577259, −1.02123321135000902402846673603, 1.87435532066942530968591045522, 2.86845563748581082415577490268, 4.86242368697793892696606577149, 6.11682495291297444976858960661, 6.65422193562869872585556181848, 7.61365351196521109246471704978, 8.898115150053610575015434221755, 9.740004347832950164245090735814, 10.93109034766312146794818989059, 11.86962544232712801783867868299

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