Properties

Label 2-845-13.10-c1-0-9
Degree $2$
Conductor $845$
Sign $-0.967 - 0.252i$
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  + (0.190 − 0.109i)2-s + (0.800 + 1.38i)3-s + (−0.975 + 1.69i)4-s + i·5-s + (0.304 + 0.175i)6-s + (0.287 + 0.166i)7-s + 0.868i·8-s + (0.219 − 0.380i)9-s + (0.109 + 0.190i)10-s + (−4.65 + 2.68i)11-s − 3.12·12-s + 0.0729·14-s + (−1.38 + 0.800i)15-s + (−1.85 − 3.21i)16-s + (−2.53 + 4.38i)17-s − 0.0965i·18-s + ⋯
L(s)  = 1  + (0.134 − 0.0776i)2-s + (0.461 + 0.800i)3-s + (−0.487 + 0.845i)4-s + 0.447i·5-s + (0.124 + 0.0717i)6-s + (0.108 + 0.0627i)7-s + 0.306i·8-s + (0.0732 − 0.126i)9-s + (0.0347 + 0.0601i)10-s + (−1.40 + 0.809i)11-s − 0.901·12-s + 0.0195·14-s + (−0.357 + 0.206i)15-s + (−0.464 − 0.803i)16-s + (−0.614 + 1.06i)17-s − 0.0227i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.967 - 0.252i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.967 - 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.155662 + 1.21240i\)
\(L(\frac12)\) \(\approx\) \(0.155662 + 1.21240i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.190 + 0.109i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.800 - 1.38i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.287 - 0.166i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.65 - 2.68i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.53 - 4.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.96 - 1.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.41 + 2.45i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.45 - 2.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.46iT - 31T^{2} \)
37 \( 1 + (-5.17 + 2.98i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.23 - 1.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.53 - 4.38i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.34iT - 47T^{2} \)
53 \( 1 + 1.56T + 53T^{2} \)
59 \( 1 + (2.34 + 1.35i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.05 - 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.94 - 5.16i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-11.0 - 6.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.68iT - 73T^{2} \)
79 \( 1 - 4.51T + 79T^{2} \)
83 \( 1 - 4.26iT - 83T^{2} \)
89 \( 1 + (-2.79 + 1.61i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.17 + 1.25i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.38300342994373651388198129742, −9.797187532723001660385424081270, −8.912554135190813089886502246957, −8.073723849054669764539181598053, −7.45141166075267481544241799595, −6.22221239196319680486448813919, −4.88319631731217711452687797384, −4.24429743522534153261940345601, −3.28013464871661020532157837796, −2.36523694123015608002082040917, 0.52878220703345017852037466947, 1.88537350915967491844950530027, 3.09668163842090058156167352864, 4.73317746548204889142991667385, 5.22440407676893868737678496809, 6.30317461378459299053206207814, 7.33377526892525332381399046544, 8.129049681119132815115010374833, 8.841882285746706545056354320614, 9.764032500340305220204705520658

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