Properties

Label 2-845-65.29-c1-0-61
Degree $2$
Conductor $845$
Sign $-0.161 + 0.986i$
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  + (2.20 − 1.27i)2-s + (1.86 − 1.07i)3-s + (2.24 − 3.88i)4-s + (0.817 + 2.08i)5-s + (2.74 − 4.74i)6-s + (−2.54 − 1.46i)7-s − 6.31i·8-s + (0.817 − 1.41i)9-s + (4.45 + 3.54i)10-s + (−0.317 − 0.550i)11-s − 9.64i·12-s − 7.48·14-s + (3.76 + 3.00i)15-s + (−3.55 − 6.16i)16-s + (1.05 + 0.611i)17-s − 4.16i·18-s + ⋯
L(s)  = 1  + (1.55 − 0.900i)2-s + (1.07 − 0.621i)3-s + (1.12 − 1.94i)4-s + (0.365 + 0.930i)5-s + (1.11 − 1.93i)6-s + (−0.961 − 0.555i)7-s − 2.23i·8-s + (0.272 − 0.472i)9-s + (1.40 + 1.12i)10-s + (−0.0957 − 0.165i)11-s − 2.78i·12-s − 1.99·14-s + (0.972 + 0.774i)15-s + (−0.889 − 1.54i)16-s + (0.257 + 0.148i)17-s − 0.981i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.161 + 0.986i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.161 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.16995 - 3.73056i\)
\(L(\frac12)\) \(\approx\) \(3.16995 - 3.73056i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.817 - 2.08i)T \)
13 \( 1 \)
good2 \( 1 + (-2.20 + 1.27i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.86 + 1.07i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (2.54 + 1.46i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.317 + 0.550i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.05 - 0.611i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.682 - 1.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.86 - 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.96T + 31T^{2} \)
37 \( 1 + (1.05 - 0.611i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.98 + 8.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.18 + 0.683i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.16iT - 47T^{2} \)
53 \( 1 + 0.642iT - 53T^{2} \)
59 \( 1 + (3.79 - 6.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.13 + 1.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.95 - 4.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.31 + 2.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + 1.03T + 79T^{2} \)
83 \( 1 - 11.8iT - 83T^{2} \)
89 \( 1 + (-6.27 - 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.8 + 7.39i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.28852790684082674198335676424, −9.459332997833743868393698332687, −8.129551417912369964646644986711, −7.02622087201009641136378106015, −6.44170452368417476222448311730, −5.50079026315482258189630996600, −4.08038631067520609227642961329, −3.21788408770049052214790001423, −2.73715843577088805535389272911, −1.63522589766857810154030954943, 2.48666145708412877044988739661, 3.25703357694838009705949320997, 4.25658390042641015876840779156, 4.95644024829652776182988458791, 5.98857433055483639310075784211, 6.61579045812212436241859786325, 7.958535797217257683290157688682, 8.517048875836219487290999979602, 9.460768998503540533392470611876, 10.08756223382850407829966110766

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