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Properties

Label 2-845-169.116-c1-0-35
Degree $2$
Conductor $845$
Sign $0.225 + 0.974i$
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  + (−1.63 − 1.84i)2-s + (0.881 + 2.32i)3-s + (−0.489 + 4.02i)4-s + (0.239 + 0.970i)5-s + (2.84 − 5.41i)6-s + (3.13 − 2.16i)7-s + (4.16 − 2.87i)8-s + (−2.37 + 2.10i)9-s + (1.39 − 2.02i)10-s + (1.99 − 2.25i)11-s + (−9.79 + 2.41i)12-s + (−0.902 − 3.49i)13-s + (−9.11 − 2.24i)14-s + (−2.04 + 1.41i)15-s + (−4.22 − 1.04i)16-s + (−4.16 − 6.02i)17-s + ⋯
L(s)  = 1  + (−1.15 − 1.30i)2-s + (0.508 + 1.34i)3-s + (−0.244 + 2.01i)4-s + (0.107 + 0.434i)5-s + (1.16 − 2.21i)6-s + (1.18 − 0.818i)7-s + (1.47 − 1.01i)8-s + (−0.792 + 0.702i)9-s + (0.442 − 0.640i)10-s + (0.603 − 0.680i)11-s + (−2.82 + 0.696i)12-s + (−0.250 − 0.968i)13-s + (−2.43 − 0.600i)14-s + (−0.528 + 0.364i)15-s + (−1.05 − 0.260i)16-s + (−1.00 − 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.225 + 0.974i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.225 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.841597 - 0.669122i\)
\(L(\frac12)\) \(\approx\) \(0.841597 - 0.669122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.239 - 0.970i)T \)
13 \( 1 + (0.902 + 3.49i)T \)
good2 \( 1 + (1.63 + 1.84i)T + (-0.241 + 1.98i)T^{2} \)
3 \( 1 + (-0.881 - 2.32i)T + (-2.24 + 1.98i)T^{2} \)
7 \( 1 + (-3.13 + 2.16i)T + (2.48 - 6.54i)T^{2} \)
11 \( 1 + (-1.99 + 2.25i)T + (-1.32 - 10.9i)T^{2} \)
17 \( 1 + (4.16 + 6.02i)T + (-6.02 + 15.8i)T^{2} \)
19 \( 1 + 7.01iT - 19T^{2} \)
23 \( 1 - 1.99T + 23T^{2} \)
29 \( 1 + (2.20 - 1.95i)T + (3.49 - 28.7i)T^{2} \)
31 \( 1 + (-2.11 + 4.02i)T + (-17.6 - 25.5i)T^{2} \)
37 \( 1 + (0.352 - 0.671i)T + (-21.0 - 30.4i)T^{2} \)
41 \( 1 + (7.52 - 2.85i)T + (30.6 - 27.1i)T^{2} \)
43 \( 1 + (-3.18 + 1.67i)T + (24.4 - 35.3i)T^{2} \)
47 \( 1 + (-7.07 + 0.859i)T + (45.6 - 11.2i)T^{2} \)
53 \( 1 + (4.15 + 6.02i)T + (-18.7 + 49.5i)T^{2} \)
59 \( 1 + (-3.36 - 13.6i)T + (-52.2 + 27.4i)T^{2} \)
61 \( 1 + (1.70 - 2.46i)T + (-21.6 - 57.0i)T^{2} \)
67 \( 1 + (-0.911 + 0.110i)T + (65.0 - 16.0i)T^{2} \)
71 \( 1 + (6.43 - 2.44i)T + (53.1 - 47.0i)T^{2} \)
73 \( 1 + (-6.23 + 7.03i)T + (-8.79 - 72.4i)T^{2} \)
79 \( 1 + (1.48 + 12.2i)T + (-76.7 + 18.9i)T^{2} \)
83 \( 1 + (1.49 + 0.567i)T + (62.1 + 55.0i)T^{2} \)
89 \( 1 - 6.81iT - 89T^{2} \)
97 \( 1 + (1.88 - 7.66i)T + (-85.8 - 45.0i)T^{2} \)
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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

−10.17331864785367852410645449610, −9.148730382093138200717368409514, −8.916067287648543194636576493457, −7.87682455743882720149014837440, −7.02676146975217696200620661439, −5.06581033292992604112044830980, −4.26327371442720167538731008205, −3.23974700058641277677379606684, −2.48045853964495863521614087178, −0.77334424608483353066478109963, 1.62281958988191594289530155204, 1.83137267621818811760949443971, 4.37776911177809112709416733290, 5.59175584248406374753160349611, 6.42438447842703116895586160441, 7.09457613762732095664075719222, 7.994508487531985397170845528053, 8.456510444436152405773948512887, 8.971630186435099265582882527461, 9.917263188644724201577645008873

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