Properties

Label 2-336-16.13-c1-0-18
Degree $2$
Conductor $336$
Sign $0.699 - 0.715i$
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  + (1.24 + 0.663i)2-s + (0.707 + 0.707i)3-s + (1.12 + 1.65i)4-s + (1.67 − 1.67i)5-s + (0.414 + 1.35i)6-s i·7-s + (0.301 + 2.81i)8-s + 1.00i·9-s + (3.21 − 0.984i)10-s + (0.497 − 0.497i)11-s + (−0.379 + 1.96i)12-s + (−3.02 − 3.02i)13-s + (0.663 − 1.24i)14-s + 2.37·15-s + (−1.48 + 3.71i)16-s − 6.75·17-s + ⋯
L(s)  = 1  + (0.883 + 0.468i)2-s + (0.408 + 0.408i)3-s + (0.560 + 0.828i)4-s + (0.751 − 0.751i)5-s + (0.169 + 0.552i)6-s − 0.377i·7-s + (0.106 + 0.994i)8-s + 0.333i·9-s + (1.01 − 0.311i)10-s + (0.149 − 0.149i)11-s + (−0.109 + 0.566i)12-s + (−0.837 − 0.837i)13-s + (0.177 − 0.333i)14-s + 0.613·15-s + (−0.372 + 0.928i)16-s − 1.63·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.699 - 0.715i$
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: \((2,\ 336,\ (\ :1/2),\ 0.699 - 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38099 + 1.00216i\)
\(L(\frac12)\) \(\approx\) \(2.38099 + 1.00216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 - 0.663i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + iT \)
good5 \( 1 + (-1.67 + 1.67i)T - 5iT^{2} \)
11 \( 1 + (-0.497 + 0.497i)T - 11iT^{2} \)
13 \( 1 + (3.02 + 3.02i)T + 13iT^{2} \)
17 \( 1 + 6.75T + 17T^{2} \)
19 \( 1 + (-2.78 - 2.78i)T + 19iT^{2} \)
23 \( 1 - 0.0553iT - 23T^{2} \)
29 \( 1 + (1.37 + 1.37i)T + 29iT^{2} \)
31 \( 1 + 0.851T + 31T^{2} \)
37 \( 1 + (2.22 - 2.22i)T - 37iT^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 + (8.33 - 8.33i)T - 43iT^{2} \)
47 \( 1 - 8.67T + 47T^{2} \)
53 \( 1 + (-5.16 + 5.16i)T - 53iT^{2} \)
59 \( 1 + (-10.6 + 10.6i)T - 59iT^{2} \)
61 \( 1 + (1.30 + 1.30i)T + 61iT^{2} \)
67 \( 1 + (-0.0901 - 0.0901i)T + 67iT^{2} \)
71 \( 1 + 5.54iT - 71T^{2} \)
73 \( 1 - 0.395iT - 73T^{2} \)
79 \( 1 - 9.36T + 79T^{2} \)
83 \( 1 + (-5.80 - 5.80i)T + 83iT^{2} \)
89 \( 1 + 11.7iT - 89T^{2} \)
97 \( 1 + 6.66T + 97T^{2} \)
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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

−11.88412267484514019498913686453, −10.78553763015290855277561898864, −9.742481542541956428202356404713, −8.781106761704144224161709769496, −7.84123688756422776393120655221, −6.74077289254698647936164739051, −5.51081256099799202652257547181, −4.79536403576899711191190533741, −3.61380184329683693393194708719, −2.19054963234387882456663301459, 2.02506042371329113269890698322, 2.72682071521651496184353921414, 4.24559009709784953764660530168, 5.47392198667159779729203284964, 6.67919400460862452737135665316, 7.08463536091685417019634606657, 8.908678913905586358046184989833, 9.681462552200374415461008864002, 10.67436708951416751878492038396, 11.59896713893777892868236669105

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