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Properties

Label 2-845-65.18-c1-0-14
Degree $2$
Conductor $845$
Sign $-0.671 + 0.740i$
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  + 2.25i·2-s + (1.40 + 1.40i)3-s − 3.06·4-s + (0.247 + 2.22i)5-s + (−3.16 + 3.16i)6-s − 1.27·7-s − 2.39i·8-s + 0.947i·9-s + (−5.00 + 0.558i)10-s + (3.86 + 3.86i)11-s + (−4.30 − 4.30i)12-s − 2.87i·14-s + (−2.77 + 3.47i)15-s − 0.731·16-s + (2.27 + 2.27i)17-s − 2.13·18-s + ⋯
L(s)  = 1  + 1.59i·2-s + (0.811 + 0.811i)3-s − 1.53·4-s + (0.110 + 0.993i)5-s + (−1.29 + 1.29i)6-s − 0.482·7-s − 0.848i·8-s + 0.315i·9-s + (−1.58 + 0.176i)10-s + (1.16 + 1.16i)11-s + (−1.24 − 1.24i)12-s − 0.768i·14-s + (−0.716 + 0.896i)15-s − 0.182·16-s + (0.552 + 0.552i)17-s − 0.502·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.671 + 0.740i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (408, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.671 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.757545 - 1.70934i\)
\(L(\frac12)\) \(\approx\) \(0.757545 - 1.70934i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.247 - 2.22i)T \)
13 \( 1 \)
good2 \( 1 - 2.25iT - 2T^{2} \)
3 \( 1 + (-1.40 - 1.40i)T + 3iT^{2} \)
7 \( 1 + 1.27T + 7T^{2} \)
11 \( 1 + (-3.86 - 3.86i)T + 11iT^{2} \)
17 \( 1 + (-2.27 - 2.27i)T + 17iT^{2} \)
19 \( 1 + (0.861 + 0.861i)T + 19iT^{2} \)
23 \( 1 + (-0.117 + 0.117i)T - 23iT^{2} \)
29 \( 1 + 9.71iT - 29T^{2} \)
31 \( 1 + (-0.233 + 0.233i)T - 31iT^{2} \)
37 \( 1 - 1.32T + 37T^{2} \)
41 \( 1 + (-0.354 + 0.354i)T - 41iT^{2} \)
43 \( 1 + (-4.71 + 4.71i)T - 43iT^{2} \)
47 \( 1 - 3.20T + 47T^{2} \)
53 \( 1 + (-4.49 - 4.49i)T + 53iT^{2} \)
59 \( 1 + (0.00162 - 0.00162i)T - 59iT^{2} \)
61 \( 1 - 1.39T + 61T^{2} \)
67 \( 1 + 6.07iT - 67T^{2} \)
71 \( 1 + (8.59 - 8.59i)T - 71iT^{2} \)
73 \( 1 - 7.34iT - 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 - 2.65T + 83T^{2} \)
89 \( 1 + (5.09 - 5.09i)T - 89iT^{2} \)
97 \( 1 + 4.18iT - 97T^{2} \)
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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

−10.16819630454064604699806686909, −9.668675517779885877691428358950, −9.009554735643501180830573444221, −8.053784142927641461329927502612, −7.20902485765780537659044775771, −6.52313570173776892692325654876, −5.80964466294977952607896781191, −4.36013897341731065664887654563, −3.81505542802421813562626320423, −2.45144387105349907422517623684, 0.896874218099087614999977722956, 1.69385448188718526518143445910, 2.96714291941155676314981331866, 3.65491274374094139543531369180, 4.84451122798141080563371889042, 6.11392585867971871707310510515, 7.27479000828896929846064660775, 8.449231433431177709437656842314, 8.949546748863224817333348081994, 9.541812266179433777719036146969

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