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Properties

Label 2-336-16.5-c1-0-23
Degree $2$
Conductor $336$
Sign $-0.762 + 0.646i$
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  + (0.919 − 1.07i)2-s + (0.707 − 0.707i)3-s + (−0.309 − 1.97i)4-s + (−1.77 − 1.77i)5-s + (−0.109 − 1.40i)6-s − i·7-s + (−2.40 − 1.48i)8-s − 1.00i·9-s + (−3.53 + 0.274i)10-s + (2.61 + 2.61i)11-s + (−1.61 − 1.17i)12-s + (−3.33 + 3.33i)13-s + (−1.07 − 0.919i)14-s − 2.50·15-s + (−3.80 + 1.22i)16-s + 5.10·17-s + ⋯
L(s)  = 1  + (0.650 − 0.759i)2-s + (0.408 − 0.408i)3-s + (−0.154 − 0.987i)4-s + (−0.792 − 0.792i)5-s + (−0.0447 − 0.575i)6-s − 0.377i·7-s + (−0.851 − 0.524i)8-s − 0.333i·9-s + (−1.11 + 0.0868i)10-s + (0.787 + 0.787i)11-s + (−0.466 − 0.340i)12-s + (−0.925 + 0.925i)13-s + (−0.287 − 0.245i)14-s − 0.647·15-s + (−0.952 + 0.305i)16-s + 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.762 + 0.646i$
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: \((2,\ 336,\ (\ :1/2),\ -0.762 + 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.600962 - 1.63880i\)
\(L(\frac12)\) \(\approx\) \(0.600962 - 1.63880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.919 + 1.07i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + iT \)
good5 \( 1 + (1.77 + 1.77i)T + 5iT^{2} \)
11 \( 1 + (-2.61 - 2.61i)T + 11iT^{2} \)
13 \( 1 + (3.33 - 3.33i)T - 13iT^{2} \)
17 \( 1 - 5.10T + 17T^{2} \)
19 \( 1 + (-0.678 + 0.678i)T - 19iT^{2} \)
23 \( 1 + 7.10iT - 23T^{2} \)
29 \( 1 + (-7.37 + 7.37i)T - 29iT^{2} \)
31 \( 1 + 0.578T + 31T^{2} \)
37 \( 1 + (-6.93 - 6.93i)T + 37iT^{2} \)
41 \( 1 + 4.09iT - 41T^{2} \)
43 \( 1 + (-3.04 - 3.04i)T + 43iT^{2} \)
47 \( 1 + 7.82T + 47T^{2} \)
53 \( 1 + (-9.06 - 9.06i)T + 53iT^{2} \)
59 \( 1 + (0.0513 + 0.0513i)T + 59iT^{2} \)
61 \( 1 + (7.18 - 7.18i)T - 61iT^{2} \)
67 \( 1 + (7.45 - 7.45i)T - 67iT^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 - 3.61iT - 73T^{2} \)
79 \( 1 + 6.07T + 79T^{2} \)
83 \( 1 + (2.13 - 2.13i)T - 83iT^{2} \)
89 \( 1 - 13.6iT - 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−11.79578387193092849323823588809, −10.22627351733785355854493596718, −9.510748681531306089832584802891, −8.465134594409016853058347932087, −7.34302135441086598211392270525, −6.29688623623019791565014052659, −4.61164915634635785437672516714, −4.21513143448547657752330788767, −2.63325084007489234998674661304, −1.06177054469726580920433795287, 3.08635767515668713350479469515, 3.54528942080708445551625572733, 5.01648349152419609915976637056, 5.99832284818398743367505597334, 7.30017756762959494196608762381, 7.86096733183225469004643711588, 8.892428069015347375845757044701, 9.983302521705491185673235621714, 11.23628356929352881763752808028, 11.95014128457594183704461678164

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