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Properties

Label 2-845-65.7-c1-0-41
Degree $2$
Conductor $845$
Sign $0.940 + 0.340i$
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  + (1.58 + 0.915i)2-s + (−0.512 − 1.91i)3-s + (0.677 + 1.17i)4-s + (1.69 − 1.45i)5-s + (0.939 − 3.50i)6-s + (1.76 + 3.06i)7-s − 1.18i·8-s + (−0.803 + 0.463i)9-s + (4.02 − 0.752i)10-s + (1.00 + 3.74i)11-s + (1.89 − 1.89i)12-s + 6.48i·14-s + (−3.65 − 2.50i)15-s + (2.43 − 4.22i)16-s + (1.95 + 0.524i)17-s − 1.69·18-s + ⋯
L(s)  = 1  + (1.12 + 0.647i)2-s + (−0.296 − 1.10i)3-s + (0.338 + 0.586i)4-s + (0.759 − 0.650i)5-s + (0.383 − 1.43i)6-s + (0.668 + 1.15i)7-s − 0.417i·8-s + (−0.267 + 0.154i)9-s + (1.27 − 0.237i)10-s + (0.302 + 1.12i)11-s + (0.548 − 0.548i)12-s + 1.73i·14-s + (−0.943 − 0.646i)15-s + (0.609 − 1.05i)16-s + (0.474 + 0.127i)17-s − 0.400·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.940 + 0.340i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.940 + 0.340i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.06915 - 0.538620i\)
\(L(\frac12)\) \(\approx\) \(3.06915 - 0.538620i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.69 + 1.45i)T \)
13 \( 1 \)
good2 \( 1 + (-1.58 - 0.915i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.512 + 1.91i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.76 - 3.06i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.00 - 3.74i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.95 - 0.524i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.518 + 0.139i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.294 + 0.0788i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.71 + 0.988i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.13 + 4.13i)T + 31iT^{2} \)
37 \( 1 + (-2.70 + 4.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.649 - 0.174i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.28 - 8.51i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 9.75T + 47T^{2} \)
53 \( 1 + (-3.16 + 3.16i)T - 53iT^{2} \)
59 \( 1 + (3.14 - 11.7i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.44 - 2.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.98 - 1.14i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.19 + 4.46i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 - 1.59iT - 79T^{2} \)
83 \( 1 - 7.57T + 83T^{2} \)
89 \( 1 + (4.54 - 1.21i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (15.4 - 8.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.924348652873880090788632466781, −9.297870130402568829817245081540, −8.152249991945685832435715572816, −7.30782033652077956986638654745, −6.41866343392385829446982987202, −5.75024934838796631770986845746, −5.10935050059026889285046138926, −4.17654878715663322820303428707, −2.35253488640686339521977504910, −1.39612149501692483280436928722, 1.66455531984593899426296289389, 3.25540819336390688373578736699, 3.75365766452158288098844516485, 4.81432045763821304972414176585, 5.40673166167496895113815774821, 6.41950503248978605406313294484, 7.60039841154012086062307504990, 8.739944427890165704839592146589, 9.864132021730974351564906183112, 10.53261787771275877583899657120

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