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Properties

Label 2-665-19.11-c1-0-35
Degree $2$
Conductor $665$
Sign $-0.915 + 0.403i$
Analytic cond. $5.31005$
Root an. cond. $2.30435$
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  + (0.745 − 1.29i)2-s + (1.62 − 2.82i)3-s + (−0.112 − 0.194i)4-s + (0.5 − 0.866i)5-s + (−2.42 − 4.20i)6-s − 7-s + 2.64·8-s + (−3.80 − 6.59i)9-s + (−0.745 − 1.29i)10-s − 4.16·11-s − 0.730·12-s + (2.44 + 4.24i)13-s + (−0.745 + 1.29i)14-s + (−1.62 − 2.82i)15-s + (2.19 − 3.80i)16-s + (−1.32 + 2.29i)17-s + ⋯
L(s)  = 1  + (0.527 − 0.913i)2-s + (0.940 − 1.62i)3-s + (−0.0560 − 0.0970i)4-s + (0.223 − 0.387i)5-s + (−0.991 − 1.71i)6-s − 0.377·7-s + 0.936·8-s + (−1.26 − 2.19i)9-s + (−0.235 − 0.408i)10-s − 1.25·11-s − 0.210·12-s + (0.679 + 1.17i)13-s + (−0.199 + 0.345i)14-s + (−0.420 − 0.728i)15-s + (0.549 − 0.952i)16-s + (−0.321 + 0.556i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(665\)    =    \(5 \cdot 7 \cdot 19\)
Sign: $-0.915 + 0.403i$
Analytic conductor: \(5.31005\)
Root analytic conductor: \(2.30435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{665} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 665,\ (\ :1/2),\ -0.915 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.566245 - 2.68881i\)
\(L(\frac12)\) \(\approx\) \(0.566245 - 2.68881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + T \)
19 \( 1 + (-4.27 + 0.875i)T \)
good2 \( 1 + (-0.745 + 1.29i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.62 + 2.82i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + 4.16T + 11T^{2} \)
13 \( 1 + (-2.44 - 4.24i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.32 - 2.29i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.11 + 1.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.65 - 4.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.565T + 31T^{2} \)
37 \( 1 - 5.73T + 37T^{2} \)
41 \( 1 + (-1.06 + 1.84i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.07 + 5.32i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.75 + 4.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.04 - 8.74i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.35 - 5.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.14 + 7.18i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.92 - 8.53i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.43 - 12.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.17 - 5.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.09 + 5.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + (-4.99 - 8.65i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.47 + 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.27836588336695200106611273087, −9.093318267255837581347657299942, −8.386811289806540105272123518222, −7.49931598268468800907520251144, −6.80095943414990803579585450576, −5.66753545292714829555951230135, −4.15567563110485461862514983084, −3.01967251429711354642267533906, −2.26783404473652878376029297595, −1.24754237334955688559499521969, 2.62986342130104868089953086044, 3.43675200694931319290719701908, 4.61757711540811727933213765344, 5.37074268846602692314519149579, 6.11501099306184418282726091543, 7.70686175548981804967795273275, 8.001173460853755351638394221863, 9.302041911217551512565245623105, 10.05892937034877592975743152435, 10.54893829780075620300505068725

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