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Properties

Label 2-368-16.13-c1-0-8
Degree $2$
Conductor $368$
Sign $0.545 - 0.838i$
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 + (−2.15 − 2.15i)3-s + (−0.931 + 1.76i)4-s + (0.480 − 0.480i)5-s + (1.03 − 4.18i)6-s + 2.39i·7-s + (−2.82 + 0.164i)8-s + 6.27i·9-s + (0.931 + 0.230i)10-s + (2.77 − 2.77i)11-s + (5.81 − 1.80i)12-s + (3.45 + 3.45i)13-s + (−2.90 + 1.75i)14-s − 2.06·15-s + (−2.26 − 3.29i)16-s + 6.32·17-s + ⋯
L(s)  = 1  + (0.516 + 0.856i)2-s + (−1.24 − 1.24i)3-s + (−0.465 + 0.884i)4-s + (0.214 − 0.214i)5-s + (0.421 − 1.70i)6-s + 0.905i·7-s + (−0.998 + 0.0582i)8-s + 2.09i·9-s + (0.294 + 0.0728i)10-s + (0.837 − 0.837i)11-s + (1.67 − 0.520i)12-s + (0.957 + 0.957i)13-s + (−0.775 + 0.468i)14-s − 0.533·15-s + (−0.565 − 0.824i)16-s + 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.545 - 0.838i$
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: \((2,\ 368,\ (\ :1/2),\ 0.545 - 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05204 + 0.570643i\)
\(L(\frac12)\) \(\approx\) \(1.05204 + 0.570643i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 + 2.15i)T + 3iT^{2} \)
5 \( 1 + (-0.480 + 0.480i)T - 5iT^{2} \)
7 \( 1 - 2.39iT - 7T^{2} \)
11 \( 1 + (-2.77 + 2.77i)T - 11iT^{2} \)
13 \( 1 + (-3.45 - 3.45i)T + 13iT^{2} \)
17 \( 1 - 6.32T + 17T^{2} \)
19 \( 1 + (-3.50 - 3.50i)T + 19iT^{2} \)
29 \( 1 + (2.24 + 2.24i)T + 29iT^{2} \)
31 \( 1 + 6.75T + 31T^{2} \)
37 \( 1 + (3.49 - 3.49i)T - 37iT^{2} \)
41 \( 1 - 9.37iT - 41T^{2} \)
43 \( 1 + (-4.56 + 4.56i)T - 43iT^{2} \)
47 \( 1 + 4.24T + 47T^{2} \)
53 \( 1 + (-6.78 + 6.78i)T - 53iT^{2} \)
59 \( 1 + (1.90 - 1.90i)T - 59iT^{2} \)
61 \( 1 + (5.55 + 5.55i)T + 61iT^{2} \)
67 \( 1 + (-7.24 - 7.24i)T + 67iT^{2} \)
71 \( 1 + 2.27iT - 71T^{2} \)
73 \( 1 + 12.4iT - 73T^{2} \)
79 \( 1 + 5.39T + 79T^{2} \)
83 \( 1 + (-2.75 - 2.75i)T + 83iT^{2} \)
89 \( 1 - 5.62iT - 89T^{2} \)
97 \( 1 - 1.10T + 97T^{2} \)
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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

−11.84669091931449072097622058212, −11.22295747458369207664932377328, −9.446957251994562387659192784224, −8.457313935866802748293992865110, −7.51053117623267288223443123429, −6.50155818616952408305648321215, −5.80878837892000965258830799972, −5.33395611335204659480611331988, −3.56642477865107408090498588831, −1.44912067532206196759076245087, 0.972083666400802943416359132554, 3.44854537667658453260635333742, 4.11939488470847156783891164903, 5.26078663991850001877357198588, 5.91159309720000789146080371022, 7.17465398653656772005904109187, 9.119236655436871050244699676186, 9.900949043723048045383249489036, 10.52579931440410396390103880687, 11.06221400797375856440121354811

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