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Properties

Label 2-299-13.10-c1-0-10
Degree $2$
Conductor $299$
Sign $0.984 - 0.174i$
Analytic cond. $2.38752$
Root an. cond. $1.54516$
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  + (−1.18 + 0.685i)2-s + (0.476 + 0.824i)3-s + (−0.0597 + 0.103i)4-s − 2.12i·5-s + (−1.13 − 0.653i)6-s + (−0.290 − 0.167i)7-s − 2.90i·8-s + (1.04 − 1.81i)9-s + (1.45 + 2.52i)10-s + (3.56 − 2.05i)11-s − 0.113·12-s + (3.40 + 1.18i)13-s + 0.460·14-s + (1.75 − 1.01i)15-s + (1.87 + 3.24i)16-s + (−1.32 + 2.29i)17-s + ⋯
L(s)  = 1  + (−0.839 + 0.484i)2-s + (0.274 + 0.476i)3-s + (−0.0298 + 0.0517i)4-s − 0.950i·5-s + (−0.461 − 0.266i)6-s + (−0.109 − 0.0634i)7-s − 1.02i·8-s + (0.348 − 0.604i)9-s + (0.460 + 0.797i)10-s + (1.07 − 0.619i)11-s − 0.0328·12-s + (0.944 + 0.329i)13-s + 0.123·14-s + (0.452 − 0.261i)15-s + (0.468 + 0.811i)16-s + (−0.321 + 0.556i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(299\)    =    \(13 \cdot 23\)
Sign: $0.984 - 0.174i$
Analytic conductor: \(2.38752\)
Root analytic conductor: \(1.54516\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{299} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 299,\ (\ :1/2),\ 0.984 - 0.174i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.921948 + 0.0808550i\)
\(L(\frac12)\) \(\approx\) \(0.921948 + 0.0808550i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.40 - 1.18i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (1.18 - 0.685i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.476 - 0.824i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.12iT - 5T^{2} \)
7 \( 1 + (0.290 + 0.167i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.56 + 2.05i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.32 - 2.29i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.88 + 2.24i)T + (9.5 + 16.4i)T^{2} \)
29 \( 1 + (1.19 + 2.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.86iT - 31T^{2} \)
37 \( 1 + (-3.34 + 1.92i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.36 + 4.25i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.17 - 5.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.81iT - 47T^{2} \)
53 \( 1 - 7.28T + 53T^{2} \)
59 \( 1 + (-2.48 - 1.43i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.71 - 4.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.35 - 4.24i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.710 - 0.409i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.32iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 + (1.95 - 1.12i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.81 - 1.04i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.75689192734410369449884292689, −10.56004860398614854396764009301, −9.444376491133319180213497800127, −8.806912358247303453404660893784, −8.484427713795130636107436695470, −6.96001277303804956521104803207, −6.13224871419826764485235669330, −4.34358440118392586025348224286, −3.67623922358020324339330423190, −1.05633621181543332505913746675, 1.52292981190264184805611954006, 2.72245126046918846019975341797, 4.39115339543401733501142289403, 6.04139820705597603002177910689, 7.02107293929652133924044779724, 8.032704669471952213625623497979, 8.976030967890611827040201622448, 9.903047481408624668127765807048, 10.76632129447130120162547530326, 11.33036627791948346665171886252

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