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Properties

Label 2-368-16.13-c1-0-6
Degree $2$
Conductor $368$
Sign $-0.978 - 0.206i$
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  + (−0.730 + 1.21i)2-s + (1.49 + 1.49i)3-s + (−0.932 − 1.76i)4-s + (−1.17 + 1.17i)5-s + (−2.89 + 0.717i)6-s + 0.836i·7-s + (2.82 + 0.163i)8-s + 1.46i·9-s + (−0.563 − 2.27i)10-s + (−2.72 + 2.72i)11-s + (1.24 − 4.03i)12-s + (3.86 + 3.86i)13-s + (−1.01 − 0.611i)14-s − 3.50·15-s + (−2.26 + 3.29i)16-s − 6.79·17-s + ⋯
L(s)  = 1  + (−0.516 + 0.856i)2-s + (0.862 + 0.862i)3-s + (−0.466 − 0.884i)4-s + (−0.524 + 0.524i)5-s + (−1.18 + 0.292i)6-s + 0.316i·7-s + (0.998 + 0.0577i)8-s + 0.487i·9-s + (−0.178 − 0.719i)10-s + (−0.823 + 0.823i)11-s + (0.360 − 1.16i)12-s + (1.07 + 1.07i)13-s + (−0.270 − 0.163i)14-s − 0.904·15-s + (−0.565 + 0.824i)16-s − 1.64·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $-0.978 - 0.206i$
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: \((2,\ 368,\ (\ :1/2),\ -0.978 - 0.206i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107394 + 1.02929i\)
\(L(\frac12)\) \(\approx\) \(0.107394 + 1.02929i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.730 - 1.21i)T \)
23 \( 1 + iT \)
good3 \( 1 + (-1.49 - 1.49i)T + 3iT^{2} \)
5 \( 1 + (1.17 - 1.17i)T - 5iT^{2} \)
7 \( 1 - 0.836iT - 7T^{2} \)
11 \( 1 + (2.72 - 2.72i)T - 11iT^{2} \)
13 \( 1 + (-3.86 - 3.86i)T + 13iT^{2} \)
17 \( 1 + 6.79T + 17T^{2} \)
19 \( 1 + (2.46 + 2.46i)T + 19iT^{2} \)
29 \( 1 + (-3.58 - 3.58i)T + 29iT^{2} \)
31 \( 1 - 5.64T + 31T^{2} \)
37 \( 1 + (-5.17 + 5.17i)T - 37iT^{2} \)
41 \( 1 + 2.36iT - 41T^{2} \)
43 \( 1 + (4.86 - 4.86i)T - 43iT^{2} \)
47 \( 1 + 6.69T + 47T^{2} \)
53 \( 1 + (-5.97 + 5.97i)T - 53iT^{2} \)
59 \( 1 + (2.79 - 2.79i)T - 59iT^{2} \)
61 \( 1 + (-8.82 - 8.82i)T + 61iT^{2} \)
67 \( 1 + (-0.659 - 0.659i)T + 67iT^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 + 9.63iT - 73T^{2} \)
79 \( 1 + 0.450T + 79T^{2} \)
83 \( 1 + (-9.95 - 9.95i)T + 83iT^{2} \)
89 \( 1 - 4.51iT - 89T^{2} \)
97 \( 1 + 2.31T + 97T^{2} \)
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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

−11.42595771280400504049348432274, −10.67242122815063784638537701519, −9.765020118920476756972814873279, −8.831903277487998403909179895114, −8.441972792723812279691101916206, −7.13450759050434410048011342002, −6.40563439695196134229038413102, −4.77345448491554585162702474386, −4.03335339351287684093780284230, −2.39511395399818928211086739556, 0.76512604202615608171771781712, 2.33977441100508406934107123901, 3.41355724765996220278334269970, 4.64175895655977580855790592459, 6.41487598456945408092999703373, 7.82293505466857752503107930543, 8.300390448304282227589421167895, 8.719091260814787657327481540430, 10.21916016720917507940416899690, 10.93411607338609696268088148796

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