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Properties

Label 2-368-16.13-c1-0-13
Degree $2$
Conductor $368$
Sign $0.0777 + 0.996i$
Analytic cond. $2.93849$
Root an. cond. $1.71420$
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  + (−1.40 − 0.111i)2-s + (−1.70 − 1.70i)3-s + (1.97 + 0.313i)4-s + (−2.61 + 2.61i)5-s + (2.21 + 2.59i)6-s + 1.66i·7-s + (−2.74 − 0.661i)8-s + 2.81i·9-s + (3.97 − 3.39i)10-s + (3.35 − 3.35i)11-s + (−2.83 − 3.90i)12-s + (1.50 + 1.50i)13-s + (0.184 − 2.34i)14-s + 8.91·15-s + (3.80 + 1.23i)16-s − 0.812·17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0786i)2-s + (−0.984 − 0.984i)3-s + (0.987 + 0.156i)4-s + (−1.16 + 1.16i)5-s + (0.904 + 1.05i)6-s + 0.628i·7-s + (−0.972 − 0.233i)8-s + 0.939i·9-s + (1.25 − 1.07i)10-s + (1.01 − 1.01i)11-s + (−0.818 − 1.12i)12-s + (0.418 + 0.418i)13-s + (0.0493 − 0.626i)14-s + 2.30·15-s + (0.950 + 0.309i)16-s − 0.196·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0777 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0777 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.0777 + 0.996i$
Analytic conductor: \(2.93849\)
Root analytic conductor: \(1.71420\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :1/2),\ 0.0777 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.294467 - 0.272392i\)
\(L(\frac12)\) \(\approx\) \(0.294467 - 0.272392i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.111i)T \)
23 \( 1 + iT \)
good3 \( 1 + (1.70 + 1.70i)T + 3iT^{2} \)
5 \( 1 + (2.61 - 2.61i)T - 5iT^{2} \)
7 \( 1 - 1.66iT - 7T^{2} \)
11 \( 1 + (-3.35 + 3.35i)T - 11iT^{2} \)
13 \( 1 + (-1.50 - 1.50i)T + 13iT^{2} \)
17 \( 1 + 0.812T + 17T^{2} \)
19 \( 1 + (3.81 + 3.81i)T + 19iT^{2} \)
29 \( 1 + (7.12 + 7.12i)T + 29iT^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 + (-3.80 + 3.80i)T - 37iT^{2} \)
41 \( 1 + 0.765iT - 41T^{2} \)
43 \( 1 + (-3.75 + 3.75i)T - 43iT^{2} \)
47 \( 1 - 2.18T + 47T^{2} \)
53 \( 1 + (1.48 - 1.48i)T - 53iT^{2} \)
59 \( 1 + (-9.46 + 9.46i)T - 59iT^{2} \)
61 \( 1 + (-0.442 - 0.442i)T + 61iT^{2} \)
67 \( 1 + (7.97 + 7.97i)T + 67iT^{2} \)
71 \( 1 + 2.88iT - 71T^{2} \)
73 \( 1 + 7.28iT - 73T^{2} \)
79 \( 1 - 1.36T + 79T^{2} \)
83 \( 1 + (-6.09 - 6.09i)T + 83iT^{2} \)
89 \( 1 - 4.98iT - 89T^{2} \)
97 \( 1 + 7.46T + 97T^{2} \)
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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

−11.36023339320395373039222929496, −10.71211308878071921448422920217, −9.198611047181941922084593091216, −8.260359593971973369233927935158, −7.34442536780783345119192970329, −6.45553625620473697462845150933, −6.12398739869297156697227726859, −3.83606431292845019629782321294, −2.40091883385074974794375894198, −0.51860020902536311059119138090, 1.12437948643308533121759092715, 3.88829739888502651118569408199, 4.56243470660021899235010289080, 5.83364991249140065131747039028, 7.04586739252778877340792431526, 8.063351235401827858795710567235, 8.933457649980123186614866089630, 9.873064775063412052567195751739, 10.61597914844362072463904116888, 11.51932899380008917920709361566

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