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Properties

Label 2-336-48.11-c1-0-22
Degree $2$
Conductor $336$
Sign $-0.230 + 0.973i$
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  + (−0.641 − 1.26i)2-s + (−1.71 − 0.262i)3-s + (−1.17 + 1.61i)4-s + (2.76 − 2.76i)5-s + (0.767 + 2.32i)6-s + 7-s + (2.79 + 0.446i)8-s + (2.86 + 0.899i)9-s + (−5.25 − 1.70i)10-s + (3.92 + 3.92i)11-s + (2.44 − 2.45i)12-s + (−1.26 + 1.26i)13-s + (−0.641 − 1.26i)14-s + (−5.45 + 4.00i)15-s + (−1.22 − 3.80i)16-s − 7.10i·17-s + ⋯
L(s)  = 1  + (−0.453 − 0.891i)2-s + (−0.988 − 0.151i)3-s + (−0.588 + 0.808i)4-s + (1.23 − 1.23i)5-s + (0.313 + 0.949i)6-s + 0.377·7-s + (0.987 + 0.158i)8-s + (0.954 + 0.299i)9-s + (−1.66 − 0.540i)10-s + (1.18 + 1.18i)11-s + (0.704 − 0.709i)12-s + (−0.351 + 0.351i)13-s + (−0.171 − 0.336i)14-s + (−1.40 + 1.03i)15-s + (−0.306 − 0.951i)16-s − 1.72i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.230 + 0.973i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.230 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.609079 - 0.770453i\)
\(L(\frac12)\) \(\approx\) \(0.609079 - 0.770453i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.641 + 1.26i)T \)
3 \( 1 + (1.71 + 0.262i)T \)
7 \( 1 - T \)
good5 \( 1 + (-2.76 + 2.76i)T - 5iT^{2} \)
11 \( 1 + (-3.92 - 3.92i)T + 11iT^{2} \)
13 \( 1 + (1.26 - 1.26i)T - 13iT^{2} \)
17 \( 1 + 7.10iT - 17T^{2} \)
19 \( 1 + (0.652 + 0.652i)T + 19iT^{2} \)
23 \( 1 + 3.98iT - 23T^{2} \)
29 \( 1 + (-0.280 - 0.280i)T + 29iT^{2} \)
31 \( 1 - 2.51iT - 31T^{2} \)
37 \( 1 + (-2.94 - 2.94i)T + 37iT^{2} \)
41 \( 1 - 5.39T + 41T^{2} \)
43 \( 1 + (1.15 - 1.15i)T - 43iT^{2} \)
47 \( 1 + 5.00T + 47T^{2} \)
53 \( 1 + (6.25 - 6.25i)T - 53iT^{2} \)
59 \( 1 + (1.75 + 1.75i)T + 59iT^{2} \)
61 \( 1 + (2.02 - 2.02i)T - 61iT^{2} \)
67 \( 1 + (0.986 + 0.986i)T + 67iT^{2} \)
71 \( 1 + 5.74iT - 71T^{2} \)
73 \( 1 - 0.913iT - 73T^{2} \)
79 \( 1 + 3.77iT - 79T^{2} \)
83 \( 1 + (-9.11 + 9.11i)T - 83iT^{2} \)
89 \( 1 - 2.10T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−11.46963586172564508618523782096, −10.24367786229975324430496893297, −9.494191916221342134979143950123, −9.021240289794490031964586076135, −7.50800900710484295040155991984, −6.39716362289575869822792812099, −4.80393999999363281445045461542, −4.66532767715144347486590540317, −2.11862367217013485240196775494, −1.08665127422949711799935256080, 1.56463492667418509015797149154, 3.85147976370041211146124086452, 5.48337928443522282050065677535, 6.13304350906310746444312380362, 6.63004233008789268575193508120, 7.86544929925434176239078325814, 9.192615729855386336369087061757, 10.00381999877392650226809211814, 10.76737290816988809724896112103, 11.34975613466097624214728455899

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