Properties

Label 2-336-16.13-c1-0-6
Degree $2$
Conductor $336$
Sign $-0.113 - 0.993i$
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  + (−0.371 + 1.36i)2-s + (−0.707 − 0.707i)3-s + (−1.72 − 1.01i)4-s + (0.116 − 0.116i)5-s + (1.22 − 0.701i)6-s + i·7-s + (2.02 − 1.97i)8-s + 1.00i·9-s + (0.116 + 0.203i)10-s + (−0.842 + 0.842i)11-s + (0.501 + 1.93i)12-s + (5.06 + 5.06i)13-s + (−1.36 − 0.371i)14-s − 0.165·15-s + (1.94 + 3.49i)16-s − 0.448·17-s + ⋯
L(s)  = 1  + (−0.262 + 0.964i)2-s + (−0.408 − 0.408i)3-s + (−0.861 − 0.507i)4-s + (0.0522 − 0.0522i)5-s + (0.501 − 0.286i)6-s + 0.377i·7-s + (0.716 − 0.698i)8-s + 0.333i·9-s + (0.0367 + 0.0642i)10-s + (−0.254 + 0.254i)11-s + (0.144 + 0.558i)12-s + (1.40 + 1.40i)13-s + (−0.364 − 0.0993i)14-s − 0.0426·15-s + (0.485 + 0.874i)16-s − 0.108·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.113 - 0.993i$
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: \((2,\ 336,\ (\ :1/2),\ -0.113 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605919 + 0.679212i\)
\(L(\frac12)\) \(\approx\) \(0.605919 + 0.679212i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.371 - 1.36i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 - iT \)
good5 \( 1 + (-0.116 + 0.116i)T - 5iT^{2} \)
11 \( 1 + (0.842 - 0.842i)T - 11iT^{2} \)
13 \( 1 + (-5.06 - 5.06i)T + 13iT^{2} \)
17 \( 1 + 0.448T + 17T^{2} \)
19 \( 1 + (-1.66 - 1.66i)T + 19iT^{2} \)
23 \( 1 - 2.66iT - 23T^{2} \)
29 \( 1 + (-0.858 - 0.858i)T + 29iT^{2} \)
31 \( 1 + 1.40T + 31T^{2} \)
37 \( 1 + (-5.76 + 5.76i)T - 37iT^{2} \)
41 \( 1 - 5.42iT - 41T^{2} \)
43 \( 1 + (-3.30 + 3.30i)T - 43iT^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 + (-7.71 + 7.71i)T - 53iT^{2} \)
59 \( 1 + (2.01 - 2.01i)T - 59iT^{2} \)
61 \( 1 + (-6.02 - 6.02i)T + 61iT^{2} \)
67 \( 1 + (3.55 + 3.55i)T + 67iT^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 - 8.39iT - 73T^{2} \)
79 \( 1 - 8.06T + 79T^{2} \)
83 \( 1 + (-7.48 - 7.48i)T + 83iT^{2} \)
89 \( 1 + 8.82iT - 89T^{2} \)
97 \( 1 + 2.19T + 97T^{2} \)
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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

−11.71347049245984195582718647001, −10.92875880509975355728018566168, −9.650742171783159555785286681150, −8.908965200120512759902366833380, −7.922959200483280645141091709917, −6.92312732136892921688991334993, −6.10430172721502785134089354724, −5.19675323822489798405164582552, −3.89759034994113051445768738810, −1.55566844776757467522553885418, 0.837906370916887306030023767241, 2.88079575416828312842986577444, 3.94981921017149395093631419050, 5.11942436902260187057474198101, 6.27511242498659024691233031911, 7.87404529511735976150211367433, 8.608737968818721539012694976781, 9.759823379520346290098741936334, 10.58320467292542600001603795334, 11.03508274863763446987361466498

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