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Properties

Label 2-845-13.12-c1-0-46
Degree $2$
Conductor $845$
Sign $-0.554 - 0.832i$
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  − 2.30i·2-s + 3-s − 3.30·4-s − i·5-s − 2.30i·6-s + i·7-s + 3.00i·8-s − 2·9-s − 2.30·10-s − 1.60i·11-s − 3.30·12-s + 2.30·14-s − i·15-s + 0.302·16-s − 7.60·17-s + 4.60i·18-s + ⋯
L(s)  = 1  − 1.62i·2-s + 0.577·3-s − 1.65·4-s − 0.447i·5-s − 0.940i·6-s + 0.377i·7-s + 1.06i·8-s − 0.666·9-s − 0.728·10-s − 0.484i·11-s − 0.953·12-s + 0.615·14-s − 0.258i·15-s + 0.0756·16-s − 1.84·17-s + 1.08i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.554 - 0.832i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.554 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.449947 + 0.840735i\)
\(L(\frac12)\) \(\approx\) \(0.449947 + 0.840735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
13 \( 1 \)
good2 \( 1 + 2.30iT - 2T^{2} \)
3 \( 1 - T + 3T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 1.60iT - 11T^{2} \)
17 \( 1 + 7.60T + 17T^{2} \)
19 \( 1 + 5.60iT - 19T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + 6.21T + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + 3.60iT - 37T^{2} \)
41 \( 1 - 3iT - 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + 9.21iT - 47T^{2} \)
53 \( 1 + 3.21T + 53T^{2} \)
59 \( 1 - 10.8iT - 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 7iT - 67T^{2} \)
71 \( 1 - 4.81iT - 71T^{2} \)
73 \( 1 - 0.788iT - 73T^{2} \)
79 \( 1 - 5.21T + 79T^{2} \)
83 \( 1 + 9.21iT - 83T^{2} \)
89 \( 1 - 6.21iT - 89T^{2} \)
97 \( 1 + 8.39iT - 97T^{2} \)
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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

−9.478373121470212273541852611688, −8.978337124948859797220329980388, −8.602386686639780786969910674099, −7.22906494741001589868184350980, −5.88406423917973855382087823922, −4.75462508972006341696082389192, −3.85044484808374319900907784407, −2.73929089197112560829296018160, −2.12010011695616190053804502294, −0.40549553316496237862697245364, 2.27669434407411454835486951992, 3.70913986405087788175847755240, 4.69568588763049536623614362532, 5.78237735914990167729217261077, 6.54615210084276495618198769249, 7.35463178769963724688243357952, 7.985501698632387911045098677466, 8.879607360961337695517342906078, 9.400906749326587168323894969927, 10.64517743373942906777073806399

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