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Properties

Label 2-845-13.9-c1-0-33
Degree $2$
Conductor $845$
Sign $0.872 + 0.488i$
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  + (−0.866 + 1.5i)2-s + (0.366 − 0.633i)3-s + (−0.5 − 0.866i)4-s − 5-s + (0.633 + 1.09i)6-s + (−1 − 1.73i)7-s − 1.73·8-s + (1.23 + 2.13i)9-s + (0.866 − 1.5i)10-s + (0.633 − 1.09i)11-s − 0.732·12-s + 3.46·14-s + (−0.366 + 0.633i)15-s + (2.49 − 4.33i)16-s + (−1.73 − 3i)17-s − 4.26·18-s + ⋯
L(s)  = 1  + (−0.612 + 1.06i)2-s + (0.211 − 0.366i)3-s + (−0.250 − 0.433i)4-s − 0.447·5-s + (0.258 + 0.448i)6-s + (−0.377 − 0.654i)7-s − 0.612·8-s + (0.410 + 0.711i)9-s + (0.273 − 0.474i)10-s + (0.191 − 0.331i)11-s − 0.211·12-s + 0.925·14-s + (−0.0945 + 0.163i)15-s + (0.624 − 1.08i)16-s + (−0.420 − 0.727i)17-s − 1.00·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.872 + 0.488i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.734187 - 0.191688i\)
\(L(\frac12)\) \(\approx\) \(0.734187 - 0.191688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
13 \( 1 \)
good2 \( 1 + (0.866 - 1.5i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.366 + 0.633i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.633 + 1.09i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.73 + 3i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.09 + 3.63i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.73 + 8.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.196T + 31T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.73 + 3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.09 + 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + (-7.56 - 13.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.19 + 10.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.19 + 12.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.633 + 1.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (0.464 - 0.803i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.875358010303033231039058012860, −9.033515436622713916300401188990, −8.201941169708284026535211248508, −7.51150989214283409769236694373, −6.96757060794251923391644345966, −6.15902672120335565290123727514, −4.91571308532941091287621851719, −3.77366062779197788746922214685, −2.45042520467653097179628716097, −0.46682104130573464460839923121, 1.36080685438362556784152299015, 2.67025620815395685920156934666, 3.59205887717867675853813012968, 4.53205595259159254271444638058, 6.06942734033980682650467207497, 6.70715516898120437988242743981, 8.227640731943373282543952092959, 8.772347981143346306713088820353, 9.599539037961078440896242090537, 10.19520777975385103019731767614

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