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Properties

Label 2-845-65.4-c1-0-34
Degree $2$
Conductor $845$
Sign $0.948 - 0.315i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  + (0.769 − 1.33i)2-s + (2.74 + 1.58i)3-s + (−0.184 − 0.319i)4-s + (−2.17 + 0.539i)5-s + (4.22 − 2.43i)6-s + (0.854 + 1.48i)7-s + 2.51·8-s + (3.52 + 6.10i)9-s + (−0.951 + 3.30i)10-s + (−2.19 − 1.26i)11-s − 1.17i·12-s + 2.63·14-s + (−6.81 − 1.95i)15-s + (2.30 − 3.98i)16-s + (−0.798 + 0.460i)17-s + 10.8·18-s + ⋯
L(s)  = 1  + (0.544 − 0.942i)2-s + (1.58 + 0.915i)3-s + (−0.0922 − 0.159i)4-s + (−0.970 + 0.241i)5-s + (1.72 − 0.995i)6-s + (0.323 + 0.559i)7-s + 0.887·8-s + (1.17 + 2.03i)9-s + (−0.300 + 1.04i)10-s + (−0.663 − 0.382i)11-s − 0.337i·12-s + 0.703·14-s + (−1.75 − 0.505i)15-s + (0.575 − 0.996i)16-s + (−0.193 + 0.111i)17-s + 2.55·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.948 - 0.315i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.948 - 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.23290 + 0.522986i\)
\(L(\frac12)\) \(\approx\) \(3.23290 + 0.522986i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.17 - 0.539i)T \)
13 \( 1 \)
good2 \( 1 + (-0.769 + 1.33i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-2.74 - 1.58i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.854 - 1.48i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.19 + 1.26i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.798 - 0.460i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.466 + 0.269i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.45 + 1.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.56 - 4.44i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.879iT - 31T^{2} \)
37 \( 1 + (-3.02 + 5.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.09 + 0.630i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.57 + 3.21i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.70T + 47T^{2} \)
53 \( 1 + 8.49iT - 53T^{2} \)
59 \( 1 + (4.08 - 2.36i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.02 + 6.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.93 + 6.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.5 - 7.24i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.95T + 73T^{2} \)
79 \( 1 + 0.496T + 79T^{2} \)
83 \( 1 - 8.63T + 83T^{2} \)
89 \( 1 + (11.1 + 6.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.95 - 5.12i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.55245654891788611290554184486, −9.407525061532446620424079500551, −8.545692546330742308396798700437, −7.957807402804364815332409202074, −7.23048753859516120434608097040, −5.27213127334344047555525326772, −4.33829821011218538974127808166, −3.65961845771962022891376227787, −2.88831883642458130732285334883, −2.11373472820963173380809945955, 1.29810993953559384803204715854, 2.64614073585809416804529625446, 3.93139863490648379919689531925, 4.57615171667701359581347102283, 6.00683857396413742359096057155, 7.10254886326970616389721874630, 7.64629635865412338618933514274, 7.934500676868366828801579296468, 8.896956371660554707322690798834, 9.944956372813751241884640937517

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