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Properties

Label 2-368-16.5-c1-0-28
Degree $2$
Conductor $368$
Sign $-0.991 + 0.130i$
Analytic cond. $2.93849$
Root an. cond. $1.71420$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (−0.633 + 0.633i)3-s + (1.73 + i)4-s + (−1 − i)5-s + (1.09 − 0.633i)6-s + 2.73i·7-s + (−1.99 − 2i)8-s + 2.19i·9-s + (1 + 1.73i)10-s + (−4.46 − 4.46i)11-s + (−1.73 + 0.464i)12-s + (2.09 − 2.09i)13-s + (1 − 3.73i)14-s + 1.26·15-s + (1.99 + 3.46i)16-s − 1.26·17-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.366 + 0.366i)3-s + (0.866 + 0.5i)4-s + (−0.447 − 0.447i)5-s + (0.448 − 0.258i)6-s + 1.03i·7-s + (−0.707 − 0.707i)8-s + 0.732i·9-s + (0.316 + 0.547i)10-s + (−1.34 − 1.34i)11-s + (−0.499 + 0.133i)12-s + (0.581 − 0.581i)13-s + (0.267 − 0.997i)14-s + 0.327·15-s + (0.499 + 0.866i)16-s − 0.307·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $-0.991 + 0.130i$
Analytic conductor: \(2.93849\)
Root analytic conductor: \(1.71420\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 368,\ (\ :1/2),\ -0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
23 \( 1 + iT \)
good3 \( 1 + (0.633 - 0.633i)T - 3iT^{2} \)
5 \( 1 + (1 + i)T + 5iT^{2} \)
7 \( 1 - 2.73iT - 7T^{2} \)
11 \( 1 + (4.46 + 4.46i)T + 11iT^{2} \)
13 \( 1 + (-2.09 + 2.09i)T - 13iT^{2} \)
17 \( 1 + 1.26T + 17T^{2} \)
19 \( 1 + (2.73 - 2.73i)T - 19iT^{2} \)
29 \( 1 + (3.09 - 3.09i)T - 29iT^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + (7.73 + 7.73i)T + 37iT^{2} \)
41 \( 1 - 3iT - 41T^{2} \)
43 \( 1 + (-1.46 - 1.46i)T + 43iT^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + (-0.196 - 0.196i)T + 53iT^{2} \)
59 \( 1 + (-10.4 - 10.4i)T + 59iT^{2} \)
61 \( 1 + (-2 + 2i)T - 61iT^{2} \)
67 \( 1 + (-0.464 + 0.464i)T - 67iT^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + 7iT - 73T^{2} \)
79 \( 1 - 2.92T + 79T^{2} \)
83 \( 1 + (-6.66 + 6.66i)T - 83iT^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−10.85307897105858936636323630191, −10.30177987036213550623997860827, −8.849662191488841616490449213421, −8.424292298328366660082524853685, −7.61348576143554426044131285646, −5.96464113274735076046139986862, −5.30200746853965537319985886805, −3.55087930145378813392451357044, −2.23242126919152895191462677501, 0, 1.88483135490906404425815753519, 3.65429588468744074944968733620, 5.19849533992952265533046248259, 6.68716037223693216751768882518, 7.09042418687421295851126375208, 7.890758913483899289665927792645, 9.136755604008863120337862852930, 10.05456586925779227816698797200, 10.89277660402553851334524050135

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