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Properties

Label 2-336-16.13-c1-0-9
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: $\chi_{336} (253, \cdot )$
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

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

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