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Properties

Label 2-845-845.392-c1-0-20
Degree $2$
Conductor $845$
Sign $-0.793 - 0.608i$
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  + (−0.392 + 0.827i)2-s + (0.706 + 0.651i)3-s + (0.733 + 0.898i)4-s + (0.686 + 2.12i)5-s + (−0.816 + 0.328i)6-s + (1.37 + 0.396i)7-s + (−2.81 + 0.693i)8-s + (−0.167 − 2.07i)9-s + (−2.03 − 0.267i)10-s + (−0.827 − 0.973i)11-s + (−0.0672 + 1.11i)12-s + (2.70 + 2.38i)13-s + (−0.866 + 0.978i)14-s + (−0.901 + 1.94i)15-s + (0.0666 − 0.326i)16-s + (1.13 + 2.06i)17-s + ⋯
L(s)  = 1  + (−0.277 + 0.585i)2-s + (0.407 + 0.376i)3-s + (0.366 + 0.449i)4-s + (0.306 + 0.951i)5-s + (−0.333 + 0.134i)6-s + (0.517 + 0.149i)7-s + (−0.994 + 0.245i)8-s + (−0.0557 − 0.690i)9-s + (−0.642 − 0.0847i)10-s + (−0.249 − 0.293i)11-s + (−0.0194 + 0.321i)12-s + (0.748 + 0.662i)13-s + (−0.231 + 0.261i)14-s + (−0.232 + 0.503i)15-s + (0.0166 − 0.0815i)16-s + (0.275 + 0.500i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.793 - 0.608i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (392, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.793 - 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.572392 + 1.68598i\)
\(L(\frac12)\) \(\approx\) \(0.572392 + 1.68598i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.686 - 2.12i)T \)
13 \( 1 + (-2.70 - 2.38i)T \)
good2 \( 1 + (0.392 - 0.827i)T + (-1.26 - 1.54i)T^{2} \)
3 \( 1 + (-0.706 - 0.651i)T + (0.241 + 2.99i)T^{2} \)
7 \( 1 + (-1.37 - 0.396i)T + (5.91 + 3.74i)T^{2} \)
11 \( 1 + (0.827 + 0.973i)T + (-1.76 + 10.8i)T^{2} \)
17 \( 1 + (-1.13 - 2.06i)T + (-9.08 + 14.3i)T^{2} \)
19 \( 1 + (0.251 - 0.0673i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.02 - 7.55i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-8.10 - 3.84i)T + (18.3 + 22.4i)T^{2} \)
31 \( 1 + (8.70 + 6.82i)T + (7.41 + 30.0i)T^{2} \)
37 \( 1 + (1.37 + 1.83i)T + (-10.2 + 35.5i)T^{2} \)
41 \( 1 + (-4.60 - 4.25i)T + (3.29 + 40.8i)T^{2} \)
43 \( 1 + (0.950 + 6.69i)T + (-41.3 + 11.9i)T^{2} \)
47 \( 1 + (-4.86 + 1.84i)T + (35.1 - 31.1i)T^{2} \)
53 \( 1 + (-3.17 + 5.24i)T + (-24.6 - 46.9i)T^{2} \)
59 \( 1 + (1.47 + 0.976i)T + (23.1 + 54.2i)T^{2} \)
61 \( 1 + (5.21 - 5.43i)T + (-2.45 - 60.9i)T^{2} \)
67 \( 1 + (-6.05 + 7.42i)T + (-13.4 - 65.6i)T^{2} \)
71 \( 1 + (1.08 - 4.81i)T + (-64.1 - 30.4i)T^{2} \)
73 \( 1 + (-0.982 + 1.42i)T + (-25.8 - 68.2i)T^{2} \)
79 \( 1 + (-4.95 + 1.88i)T + (59.1 - 52.3i)T^{2} \)
83 \( 1 + (0.468 + 0.892i)T + (-47.1 + 68.3i)T^{2} \)
89 \( 1 + (15.9 + 4.26i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.1 + 4.06i)T + (77.5 - 58.2i)T^{2} \)
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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

−10.48442278445554060292152498197, −9.464447877129547014790040602358, −8.792564028940176232858199494425, −7.961881391067159175383098598551, −7.15051714236191399988780341756, −6.29036281137222468719554115834, −5.61813158951595799749685586951, −3.86182592660221694888714508088, −3.25771478172992279603953019147, −2.00417052155555598267533397598, 0.927315947309341607574716388767, 1.94549309986740631335119476580, 2.89967881613987345037064480211, 4.53442663875842432871688838720, 5.38314810510218526256482012092, 6.31751109023713820421623709944, 7.51070360520280134551401988546, 8.372820227494876550249511243233, 8.926452464125874279524407725784, 10.03858848326051801694857208703

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