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Properties

Label 2-845-65.57-c1-0-63
Degree $2$
Conductor $845$
Sign $-0.668 + 0.743i$
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  + 1.51·2-s + (0.478 − 0.478i)3-s + 0.304·4-s + (−0.600 − 2.15i)5-s + (0.726 − 0.726i)6-s − 2.59i·7-s − 2.57·8-s + 2.54i·9-s + (−0.911 − 3.26i)10-s + (−3.53 − 3.53i)11-s + (0.145 − 0.145i)12-s − 3.93i·14-s + (−1.31 − 0.743i)15-s − 4.51·16-s + (0.0578 − 0.0578i)17-s + 3.85i·18-s + ⋯
L(s)  = 1  + 1.07·2-s + (0.276 − 0.276i)3-s + 0.152·4-s + (−0.268 − 0.963i)5-s + (0.296 − 0.296i)6-s − 0.980i·7-s − 0.910·8-s + 0.847i·9-s + (−0.288 − 1.03i)10-s + (−1.06 − 1.06i)11-s + (0.0420 − 0.0420i)12-s − 1.05i·14-s + (−0.340 − 0.191i)15-s − 1.12·16-s + (0.0140 − 0.0140i)17-s + 0.909i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.668 + 0.743i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.668 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.715285 - 1.60537i\)
\(L(\frac12)\) \(\approx\) \(0.715285 - 1.60537i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.600 + 2.15i)T \)
13 \( 1 \)
good2 \( 1 - 1.51T + 2T^{2} \)
3 \( 1 + (-0.478 + 0.478i)T - 3iT^{2} \)
7 \( 1 + 2.59iT - 7T^{2} \)
11 \( 1 + (3.53 + 3.53i)T + 11iT^{2} \)
17 \( 1 + (-0.0578 + 0.0578i)T - 17iT^{2} \)
19 \( 1 + (-1.98 - 1.98i)T + 19iT^{2} \)
23 \( 1 + (2.86 + 2.86i)T + 23iT^{2} \)
29 \( 1 + 4.98iT - 29T^{2} \)
31 \( 1 + (-2.32 + 2.32i)T - 31iT^{2} \)
37 \( 1 + 0.571iT - 37T^{2} \)
41 \( 1 + (-7.36 + 7.36i)T - 41iT^{2} \)
43 \( 1 + (0.0967 + 0.0967i)T + 43iT^{2} \)
47 \( 1 - 2.30iT - 47T^{2} \)
53 \( 1 + (-6.70 + 6.70i)T - 53iT^{2} \)
59 \( 1 + (1.89 - 1.89i)T - 59iT^{2} \)
61 \( 1 - 5.48T + 61T^{2} \)
67 \( 1 + 15.7T + 67T^{2} \)
71 \( 1 + (-5.43 + 5.43i)T - 71iT^{2} \)
73 \( 1 - 6.61T + 73T^{2} \)
79 \( 1 - 5.71iT - 79T^{2} \)
83 \( 1 - 3.70iT - 83T^{2} \)
89 \( 1 + (-12.6 + 12.6i)T - 89iT^{2} \)
97 \( 1 + 5.36T + 97T^{2} \)
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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

−9.988251019961641610779110977956, −8.820316403529235880058790459770, −8.085790474275617732407918181863, −7.47980371683897613549179645795, −6.02481553795834667328241650756, −5.31466632265821676703340711542, −4.45318407688357791948912893327, −3.67344887305832739901946803443, −2.43403345527550292463894829431, −0.56497595716273295532171685868, 2.49631620244194484898100354582, 3.12798968074973759409898268466, 4.15013874568784261323081315470, 5.12513765229706922332613994952, 5.97345559575620777282705862665, 6.85869379273956848452147800182, 7.84744049224544331008424469092, 8.970633291892205945685200691602, 9.644215204735738198368414913535, 10.50071591249245146926751239454

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