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Properties

Label 2-336-21.20-c3-0-0
Degree $2$
Conductor $336$
Sign $-0.149 - 0.988i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.04 − 1.22i)3-s − 10.0·5-s + (−7 − 17.1i)7-s + (23.9 + 12.3i)9-s − 32.9i·11-s − 56.3i·13-s + (50.9 + 12.3i)15-s − 60.5·17-s − 36.7i·19-s + (14.3 + 95.1i)21-s + 90.7i·23-s − 23.0·25-s + (−106. − 91.8i)27-s + 57.7i·29-s + 254. i·31-s + ⋯
L(s)  = 1  + (−0.971 − 0.235i)3-s − 0.903·5-s + (−0.377 − 0.925i)7-s + (0.888 + 0.458i)9-s − 0.904i·11-s − 1.20i·13-s + (0.877 + 0.212i)15-s − 0.864·17-s − 0.443i·19-s + (0.149 + 0.988i)21-s + 0.822i·23-s − 0.184·25-s + (−0.755 − 0.654i)27-s + 0.369i·29-s + 1.47i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 - 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.149 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.149 - 0.988i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.149 - 0.988i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1312525028\)
\(L(\frac12)\) \(\approx\) \(0.1312525028\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.04 + 1.22i)T \)
7 \( 1 + (7 + 17.1i)T \)
good5 \( 1 + 10.0T + 125T^{2} \)
11 \( 1 + 32.9iT - 1.33e3T^{2} \)
13 \( 1 + 56.3iT - 2.19e3T^{2} \)
17 \( 1 + 60.5T + 4.91e3T^{2} \)
19 \( 1 + 36.7iT - 6.85e3T^{2} \)
23 \( 1 - 90.7iT - 1.21e4T^{2} \)
29 \( 1 - 57.7iT - 2.43e4T^{2} \)
31 \( 1 - 254. iT - 2.97e4T^{2} \)
37 \( 1 - 230T + 5.06e4T^{2} \)
41 \( 1 - 141.T + 6.89e4T^{2} \)
43 \( 1 + 44T + 7.95e4T^{2} \)
47 \( 1 + 343.T + 1.03e5T^{2} \)
53 \( 1 - 206. iT - 1.48e5T^{2} \)
59 \( 1 - 131.T + 2.05e5T^{2} \)
61 \( 1 - 71.0iT - 2.26e5T^{2} \)
67 \( 1 - 64T + 3.00e5T^{2} \)
71 \( 1 - 461. iT - 3.57e5T^{2} \)
73 \( 1 - 88.1iT - 3.89e5T^{2} \)
79 \( 1 - 442T + 4.93e5T^{2} \)
83 \( 1 + 494.T + 5.71e5T^{2} \)
89 \( 1 - 484.T + 7.04e5T^{2} \)
97 \( 1 - 1.09e3iT - 9.12e5T^{2} \)
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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

−11.13835943927760902175538832428, −10.85449749579816889413204002873, −9.753144389789363465634060116459, −8.315226601086125125903409570506, −7.46318018103203798544884164988, −6.61881429271239653562791726818, −5.50519107520707580597242807305, −4.34361967312354276068741262228, −3.24520142985374756096313583801, −0.961147966844437532295884607482, 0.06830889117169383686571450782, 2.12628912443537593317918885320, 3.99834521209472546735275099108, 4.70204715008613220875844922955, 6.06583023160083045286257012081, 6.81353233739367418091863392493, 7.952162311326777304000185504087, 9.214926731535977121564611484131, 9.893787780631222385043140636655, 11.19599611321630655208275261481

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