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Properties

Label 2-374-11.5-c1-0-1
Degree $2$
Conductor $374$
Sign $-0.101 - 0.994i$
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  + (−0.309 − 0.951i)2-s + (0.864 + 0.628i)3-s + (−0.809 + 0.587i)4-s + (−0.294 + 0.906i)5-s + (0.330 − 1.01i)6-s + (−3.77 + 2.73i)7-s + (0.809 + 0.587i)8-s + (−0.574 − 1.76i)9-s + 0.953·10-s + (−2.36 + 2.32i)11-s − 1.06·12-s + (1.17 + 3.60i)13-s + (3.77 + 2.73i)14-s + (−0.824 + 0.598i)15-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.499 + 0.362i)3-s + (−0.404 + 0.293i)4-s + (−0.131 + 0.405i)5-s + (0.134 − 0.414i)6-s + (−1.42 + 1.03i)7-s + (0.286 + 0.207i)8-s + (−0.191 − 0.589i)9-s + 0.301·10-s + (−0.714 + 0.700i)11-s − 0.308·12-s + (0.324 + 0.999i)13-s + (1.00 + 0.732i)14-s + (−0.212 + 0.154i)15-s + (0.0772 − 0.237i)16-s + (−0.0749 + 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 374 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(374\)    =    \(2 \cdot 11 \cdot 17\)
Sign: $-0.101 - 0.994i$
Analytic conductor: \(2.98640\)
Root analytic conductor: \(1.72812\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{374} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 374,\ (\ :1/2),\ -0.101 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.504889 + 0.558923i\)
\(L(\frac12)\) \(\approx\) \(0.504889 + 0.558923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (2.36 - 2.32i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (-0.864 - 0.628i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.294 - 0.906i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (3.77 - 2.73i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.17 - 3.60i)T + (-10.5 + 7.64i)T^{2} \)
19 \( 1 + (0.744 + 0.541i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 7.66T + 23T^{2} \)
29 \( 1 + (-2.25 + 1.64i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.43 - 4.41i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.366 - 0.266i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.75 - 5.63i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.302T + 43T^{2} \)
47 \( 1 + (-3.59 - 2.61i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.73 + 5.33i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.58 + 4.78i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.29 - 13.2i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 4.07T + 67T^{2} \)
71 \( 1 + (-3.30 + 10.1i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.04 - 4.39i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.07 - 3.31i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.90 - 5.86i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + (4.33 + 13.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.75888191731138700099035589499, −10.50249385350014156007984918794, −9.695561580186681595327360118778, −9.160843619702812276776320127565, −8.266542690017854345130593971313, −6.82995941721085104159882383834, −5.94237179916219438637181545553, −4.29350565975176607710947696145, −3.21973389692345123906414072184, −2.36959359081312373501063338021, 0.49162577646795967062615664750, 2.82670091842327053794939265756, 4.05247851607060251946148273140, 5.51864019411063074955178171626, 6.43640861656677776068758388574, 7.59690554097899994362142687840, 8.106774102841191792986977005054, 9.100016646853592382864130535602, 10.25538429661832826800424915331, 10.70337944538450421768408851348

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