Properties

Label 2-336-16.5-c1-0-7
Degree $2$
Conductor $336$
Sign $-0.0328 - 0.999i$
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  + (0.421 + 1.34i)2-s + (0.707 − 0.707i)3-s + (−1.64 + 1.13i)4-s + (2.44 + 2.44i)5-s + (1.25 + 0.656i)6-s i·7-s + (−2.22 − 1.73i)8-s − 1.00i·9-s + (−2.26 + 4.33i)10-s + (2.14 + 2.14i)11-s + (−0.357 + 1.96i)12-s + (−3.07 + 3.07i)13-s + (1.34 − 0.421i)14-s + 3.45·15-s + (1.40 − 3.74i)16-s + 1.91·17-s + ⋯
L(s)  = 1  + (0.298 + 0.954i)2-s + (0.408 − 0.408i)3-s + (−0.822 + 0.569i)4-s + (1.09 + 1.09i)5-s + (0.511 + 0.267i)6-s − 0.377i·7-s + (−0.788 − 0.615i)8-s − 0.333i·9-s + (−0.717 + 1.36i)10-s + (0.645 + 0.645i)11-s + (−0.103 + 0.568i)12-s + (−0.852 + 0.852i)13-s + (0.360 − 0.112i)14-s + 0.892·15-s + (0.352 − 0.935i)16-s + 0.463·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0328 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0328 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.0328 - 0.999i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.0328 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28044 + 1.32321i\)
\(L(\frac12)\) \(\approx\) \(1.28044 + 1.32321i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.421 - 1.34i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + iT \)
good5 \( 1 + (-2.44 - 2.44i)T + 5iT^{2} \)
11 \( 1 + (-2.14 - 2.14i)T + 11iT^{2} \)
13 \( 1 + (3.07 - 3.07i)T - 13iT^{2} \)
17 \( 1 - 1.91T + 17T^{2} \)
19 \( 1 + (-1.94 + 1.94i)T - 19iT^{2} \)
23 \( 1 + 5.41iT - 23T^{2} \)
29 \( 1 + (4.71 - 4.71i)T - 29iT^{2} \)
31 \( 1 + 4.00T + 31T^{2} \)
37 \( 1 + (-0.294 - 0.294i)T + 37iT^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (-3.10 - 3.10i)T + 43iT^{2} \)
47 \( 1 - 7.75T + 47T^{2} \)
53 \( 1 + (3.04 + 3.04i)T + 53iT^{2} \)
59 \( 1 + (9.93 + 9.93i)T + 59iT^{2} \)
61 \( 1 + (-6.60 + 6.60i)T - 61iT^{2} \)
67 \( 1 + (-7.34 + 7.34i)T - 67iT^{2} \)
71 \( 1 + 6.77iT - 71T^{2} \)
73 \( 1 - 2.35iT - 73T^{2} \)
79 \( 1 + 9.34T + 79T^{2} \)
83 \( 1 + (-11.5 + 11.5i)T - 83iT^{2} \)
89 \( 1 + 3.57iT - 89T^{2} \)
97 \( 1 + 5.98T + 97T^{2} \)
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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

−12.10079121292566182217734594706, −10.70885053663430227687327801174, −9.591275966250547796725762194699, −9.115379450896087765288660631618, −7.54934654883547230036124764559, −6.96004429582413169120439506152, −6.29149527338444651063547583772, −5.00217869995138151124187190637, −3.60239810041980465160136321900, −2.20046656517100582882972049846, 1.37586803146084657513373169904, 2.75442141803661846539260886189, 4.05948047778441277994255009726, 5.41385545136135178816958199997, 5.71378965865158224698960850637, 7.902720341845153411092104213708, 9.008833598503471013683033693472, 9.513672026991046301631761508419, 10.17640057356635827577290663461, 11.40551691885353570887537038144

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