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Properties

Label 2-374-11.4-c1-0-9
Degree $2$
Conductor $374$
Sign $0.530 + 0.847i$
Analytic cond. $2.98640$
Root an. cond. $1.72812$
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  + (−0.809 − 0.587i)2-s + (0.429 − 1.32i)3-s + (0.309 + 0.951i)4-s + (1.85 − 1.34i)5-s + (−1.12 + 0.817i)6-s + (0.942 + 2.90i)7-s + (0.309 − 0.951i)8-s + (0.863 + 0.627i)9-s − 2.29·10-s + (0.309 − 3.30i)11-s + 1.39·12-s + (5.34 + 3.88i)13-s + (0.942 − 2.90i)14-s + (−0.985 − 3.03i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.248 − 0.763i)3-s + (0.154 + 0.475i)4-s + (0.830 − 0.603i)5-s + (−0.459 + 0.333i)6-s + (0.356 + 1.09i)7-s + (0.109 − 0.336i)8-s + (0.287 + 0.209i)9-s − 0.725·10-s + (0.0931 − 0.995i)11-s + 0.401·12-s + (1.48 + 1.07i)13-s + (0.251 − 0.775i)14-s + (−0.254 − 0.783i)15-s + (−0.202 + 0.146i)16-s + (−0.196 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 374 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(374\)    =    \(2 \cdot 11 \cdot 17\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(2.98640\)
Root analytic conductor: \(1.72812\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{374} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 374,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22132 - 0.676769i\)
\(L(\frac12)\) \(\approx\) \(1.22132 - 0.676769i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.309 + 3.30i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
good3 \( 1 + (-0.429 + 1.32i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.85 + 1.34i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.942 - 2.90i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-5.34 - 3.88i)T + (4.01 + 12.3i)T^{2} \)
19 \( 1 + (-0.0488 + 0.150i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 7.94T + 23T^{2} \)
29 \( 1 + (1.93 + 5.95i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.11 + 0.809i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.68 + 8.27i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.33 - 7.18i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.25T + 43T^{2} \)
47 \( 1 + (-1.77 + 5.47i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.87 + 1.35i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.698 - 2.14i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.61 - 3.34i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.53T + 67T^{2} \)
71 \( 1 + (12.2 - 8.90i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.23 - 13.0i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (11.6 + 8.44i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.5 + 7.64i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.57T + 89T^{2} \)
97 \( 1 + (-3.17 - 2.30i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.41934411854539442105418296292, −10.24893172547183085222906172209, −9.087192501923558972466086714404, −8.672726842439180856451389796668, −7.81663300377865254747217716839, −6.35629220345614973391262096136, −5.69568600243164151895838654328, −4.01648946923665165162246099727, −2.24127718975178100684997857127, −1.49709257577270784392572887495, 1.57708054583562620862334466561, 3.45260113001573471554510767425, 4.56448025169199764498573848118, 5.92081404357743250984557654464, 6.85348150686694695914599985229, 7.79352289765152822140224182906, 8.877363591389766658189718902643, 9.921975213592566942232973522185, 10.36310610407145085075959057133, 10.91118667214868669598432002598

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