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Properties

Label 2-845-65.2-c1-0-53
Degree $2$
Conductor $845$
Sign $0.659 + 0.751i$
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  + (0.246 + 0.427i)2-s + (0.908 + 0.243i)3-s + (0.878 − 1.52i)4-s + (2.21 − 0.284i)5-s + (0.120 + 0.448i)6-s + (−3.18 − 1.83i)7-s + 1.85·8-s + (−1.83 − 1.05i)9-s + (0.669 + 0.878i)10-s + (0.177 − 0.664i)11-s + (1.16 − 1.16i)12-s − 1.81i·14-s + (2.08 + 0.281i)15-s + (−1.29 − 2.24i)16-s + (0.614 + 2.29i)17-s − 1.04i·18-s + ⋯
L(s)  = 1  + (0.174 + 0.302i)2-s + (0.524 + 0.140i)3-s + (0.439 − 0.760i)4-s + (0.991 − 0.127i)5-s + (0.0490 + 0.183i)6-s + (−1.20 − 0.694i)7-s + 0.655·8-s + (−0.610 − 0.352i)9-s + (0.211 + 0.277i)10-s + (0.0536 − 0.200i)11-s + (0.337 − 0.337i)12-s − 0.485i·14-s + (0.538 + 0.0726i)15-s + (−0.324 − 0.562i)16-s + (0.149 + 0.556i)17-s − 0.246i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.659 + 0.751i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.659 + 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09241 - 0.948340i\)
\(L(\frac12)\) \(\approx\) \(2.09241 - 0.948340i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.21 + 0.284i)T \)
13 \( 1 \)
good2 \( 1 + (-0.246 - 0.427i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.908 - 0.243i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (3.18 + 1.83i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.177 + 0.664i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.614 - 2.29i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-5.29 + 1.41i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.350 + 1.30i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-8.24 + 4.75i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.81 - 4.81i)T - 31iT^{2} \)
37 \( 1 + (1.58 - 0.917i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.534 - 0.143i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.09 - 0.560i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 3.80iT - 47T^{2} \)
53 \( 1 + (2.47 - 2.47i)T - 53iT^{2} \)
59 \( 1 + (2.69 + 10.0i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.09 - 5.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.12 - 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.73 - 6.47i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 3.37T + 73T^{2} \)
79 \( 1 - 3.12iT - 79T^{2} \)
83 \( 1 - 2.13iT - 83T^{2} \)
89 \( 1 + (-3.26 - 0.874i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.53 + 6.12i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.955285949746645028373796542079, −9.461384011030403901841137861844, −8.535089679118406133972174784342, −7.27012573786140360814496302296, −6.42913022675565511219989367332, −5.93053406437799495310551816150, −4.89863075745007563579266592800, −3.47214347753233444809079878108, −2.56284746214491979484427029938, −1.01969916957812235343399182345, 1.93290957803353920335864744863, 2.90970569768107473376049398212, 3.33200059264854442046286386785, 5.06986644291770976462679231703, 5.99065080180027095452482020828, 6.88801478677472196352642805740, 7.72562959871180684557206864955, 8.797804974002115465450165766022, 9.397797881846258869942244118608, 10.21471997997349345648809398293

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