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Properties

Label 2-665-19.7-c1-0-25
Degree $2$
Conductor $665$
Sign $0.999 - 0.0375i$
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  + (0.302 + 0.523i)2-s + (0.354 + 0.614i)3-s + (0.817 − 1.41i)4-s + (0.5 + 0.866i)5-s + (−0.214 + 0.371i)6-s − 7-s + 2.19·8-s + (1.24 − 2.16i)9-s + (−0.302 + 0.523i)10-s − 0.340·11-s + 1.15·12-s + (3.17 − 5.49i)13-s + (−0.302 − 0.523i)14-s + (−0.354 + 0.614i)15-s + (−0.969 − 1.67i)16-s + (2.54 + 4.41i)17-s + ⋯
L(s)  = 1  + (0.213 + 0.370i)2-s + (0.204 + 0.354i)3-s + (0.408 − 0.707i)4-s + (0.223 + 0.387i)5-s + (−0.0875 + 0.151i)6-s − 0.377·7-s + 0.777·8-s + (0.416 − 0.720i)9-s + (−0.0956 + 0.165i)10-s − 0.102·11-s + 0.334·12-s + (0.879 − 1.52i)13-s + (−0.0808 − 0.139i)14-s + (−0.0915 + 0.158i)15-s + (−0.242 − 0.419i)16-s + (0.618 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(665\)    =    \(5 \cdot 7 \cdot 19\)
Sign: $0.999 - 0.0375i$
Analytic conductor: \(5.31005\)
Root analytic conductor: \(2.30435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{665} (596, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 665,\ (\ :1/2),\ 0.999 - 0.0375i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14762 + 0.0403443i\)
\(L(\frac12)\) \(\approx\) \(2.14762 + 0.0403443i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.22 - 1.08i)T \)
good2 \( 1 + (-0.302 - 0.523i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.354 - 0.614i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + 0.340T + 11T^{2} \)
13 \( 1 + (-3.17 + 5.49i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.54 - 4.41i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.28 + 3.95i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.46 - 7.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.98T + 31T^{2} \)
37 \( 1 + 5.09T + 37T^{2} \)
41 \( 1 + (-5.15 - 8.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.54 + 2.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.43 + 5.94i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.62 - 6.28i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.28 + 10.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.62 - 8.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.37 + 2.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.65 - 2.86i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.01 - 6.94i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.89 + 3.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.93T + 83T^{2} \)
89 \( 1 + (8.59 - 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.81 - 3.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.50600121850357771497726585850, −9.922391225117823905525177631498, −8.797610185222131381213749753046, −7.86270833804282339900361019643, −6.65351560420198975556681616470, −6.17603827827216047495803920902, −5.24652873657744941649466390605, −3.92929924419163782390043988092, −2.92389515730358385652827210690, −1.25235284977949843713114010027, 1.63181214127192164033378758206, 2.60899021746538729959700511927, 3.90572955083752318576605638144, 4.75482397172763060754111902012, 6.18239824084027760009457030798, 7.09344646910155609624498347766, 7.80816211374419249522723350421, 8.780856771309166145527580194621, 9.599423428870587885710224796719, 10.71254161288893698242392280649

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