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Properties

Label 2-845-65.4-c1-0-37
Degree $2$
Conductor $845$
Sign $0.902 + 0.430i$
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  + (−0.593 + 1.02i)2-s + (0.298 + 0.172i)3-s + (0.295 + 0.511i)4-s + (−1.71 − 1.44i)5-s + (−0.354 + 0.204i)6-s + (1.01 + 1.75i)7-s − 3.07·8-s + (−1.44 − 2.49i)9-s + (2.49 − 0.903i)10-s + (−3.36 − 1.94i)11-s + 0.203i·12-s − 2.40·14-s + (−0.262 − 0.725i)15-s + (1.23 − 2.14i)16-s + (4.71 − 2.72i)17-s + 3.42·18-s + ⋯
L(s)  = 1  + (−0.419 + 0.727i)2-s + (0.172 + 0.0996i)3-s + (0.147 + 0.255i)4-s + (−0.764 − 0.644i)5-s + (−0.144 + 0.0836i)6-s + (0.383 + 0.664i)7-s − 1.08·8-s + (−0.480 − 0.831i)9-s + (0.789 − 0.285i)10-s + (−1.01 − 0.585i)11-s + 0.0588i·12-s − 0.644·14-s + (−0.0678 − 0.187i)15-s + (0.308 − 0.535i)16-s + (1.14 − 0.660i)17-s + 0.806·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.902 + 0.430i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.902 + 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.849809 - 0.192075i\)
\(L(\frac12)\) \(\approx\) \(0.849809 - 0.192075i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.71 + 1.44i)T \)
13 \( 1 \)
good2 \( 1 + (0.593 - 1.02i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.298 - 0.172i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.01 - 1.75i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.36 + 1.94i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-4.71 + 2.72i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.09 + 2.94i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.298 - 0.172i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.18iT - 31T^{2} \)
37 \( 1 + (-2.72 + 4.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.156 - 0.0902i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.15 - 0.669i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + 2.42iT - 53T^{2} \)
59 \( 1 + (-6.11 + 3.53i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.38 + 5.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.20 + 3.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.62 + 0.940i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.86T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 7.83T + 83T^{2} \)
89 \( 1 + (-10.6 - 6.12i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.90 - 5.02i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.676663237848734825265567978941, −9.109657137895322666550783275019, −8.153329479693036264262865122583, −7.941077802382479216060550055315, −6.89102449682547636992196785446, −5.69637970749947242306174978041, −5.10434586987974987949461607347, −3.51919723766488300497462739925, −2.80573400346907756921409052316, −0.50589849786008627674678486504, 1.35574753455535240868419097884, 2.66727120306995862356919597115, 3.46284567545541437498118560659, 4.87792588828424371887471214899, 5.82191569831399487282023156462, 7.11714081124122201364240883814, 7.81975407976115465490737786302, 8.398511320800135092756756040170, 9.872878198834180428241954802355, 10.30735161182016955641176257936

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