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Properties

Label 2-336-112.101-c2-0-41
Degree $2$
Conductor $336$
Sign $-0.607 + 0.794i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.57i)2-s + (1.67 − 0.448i)3-s + (−0.982 + 3.87i)4-s + (−0.954 − 0.255i)5-s + (−2.76 − 2.09i)6-s + (−2.53 − 6.52i)7-s + (7.32 − 3.21i)8-s + (2.59 − 1.50i)9-s + (0.769 + 1.82i)10-s + (10.8 − 2.90i)11-s + (0.0935 + 6.92i)12-s + (10.5 + 10.5i)13-s + (−7.17 + 12.0i)14-s − 1.71·15-s + (−14.0 − 7.62i)16-s + (−24.5 − 14.1i)17-s + ⋯
L(s)  = 1  + (−0.614 − 0.789i)2-s + (0.557 − 0.149i)3-s + (−0.245 + 0.969i)4-s + (−0.190 − 0.0511i)5-s + (−0.460 − 0.348i)6-s + (−0.362 − 0.931i)7-s + (0.915 − 0.401i)8-s + (0.288 − 0.166i)9-s + (0.0769 + 0.182i)10-s + (0.986 − 0.264i)11-s + (0.00779 + 0.577i)12-s + (0.814 + 0.814i)13-s + (−0.512 + 0.858i)14-s − 0.114·15-s + (−0.879 − 0.476i)16-s + (−1.44 − 0.833i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.607 + 0.794i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ -0.607 + 0.794i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.543583 - 1.10037i\)
\(L(\frac12)\) \(\approx\) \(0.543583 - 1.10037i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 1.57i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
7 \( 1 + (2.53 + 6.52i)T \)
good5 \( 1 + (0.954 + 0.255i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-10.8 + 2.90i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-10.5 - 10.5i)T + 169iT^{2} \)
17 \( 1 + (24.5 + 14.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (0.443 - 1.65i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-25.3 + 14.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-25.9 + 25.9i)T - 841iT^{2} \)
31 \( 1 + (36.0 + 20.7i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-7.08 + 26.4i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 7.47T + 1.68e3T^{2} \)
43 \( 1 + (60.4 + 60.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-49.7 + 28.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (12.3 - 3.29i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-10.0 - 37.3i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-14.3 + 53.3i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-32.8 - 122. i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 7.80iT - 5.04e3T^{2} \)
73 \( 1 + (15.7 - 27.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-55.2 - 95.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-32.7 - 32.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (15.1 + 26.2i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 80.2iT - 9.40e3T^{2} \)
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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

−11.10207170724838030040570643581, −10.00928150250907257729000193735, −9.081739259959323430991352151314, −8.554981557799450030163093051933, −7.24486833051579020722821961653, −6.60914442154146997402285605057, −4.30814338116750515274504021212, −3.71995670007150486938468850906, −2.21861692275628689969868294022, −0.69252515606800528801656792536, 1.63631560977074862418599957957, 3.39295995787451521860404704219, 4.83691501046032711712216395121, 6.08441363843874910705277558861, 6.86302241731898265432160215898, 8.090492676028094458758377274848, 8.903289568858866006517524113690, 9.350285209862569777877937621099, 10.57407021836035995697079499556, 11.40505894782726590895442490206

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