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Properties

Label 2-845-65.58-c1-0-61
Degree $2$
Conductor $845$
Sign $0.756 + 0.653i$
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 + (1.91 − 0.512i)3-s + (0.677 + 1.17i)4-s + (−1.69 − 1.45i)5-s + (3.50 + 0.939i)6-s + (−1.76 − 3.06i)7-s − 1.18i·8-s + (0.803 − 0.463i)9-s + (−1.36 − 3.86i)10-s + (3.74 − 1.00i)11-s + (1.89 + 1.89i)12-s − 6.48i·14-s + (−3.99 − 1.91i)15-s + (2.43 − 4.22i)16-s + (−0.524 + 1.95i)17-s + 1.69·18-s + ⋯
L(s)  = 1  + (1.12 + 0.647i)2-s + (1.10 − 0.296i)3-s + (0.338 + 0.586i)4-s + (−0.759 − 0.650i)5-s + (1.43 + 0.383i)6-s + (−0.668 − 1.15i)7-s − 0.417i·8-s + (0.267 − 0.154i)9-s + (−0.430 − 1.22i)10-s + (1.12 − 0.302i)11-s + (0.548 + 0.548i)12-s − 1.73i·14-s + (−1.03 − 0.494i)15-s + (0.609 − 1.05i)16-s + (−0.127 + 0.474i)17-s + 0.400·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.756 + 0.653i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.756 + 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.03108 - 1.12807i\)
\(L(\frac12)\) \(\approx\) \(3.03108 - 1.12807i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.91 + 0.512i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.76 + 3.06i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.74 + 1.00i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.524 - 1.95i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.139 - 0.518i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.0788 + 0.294i)T + (-19.9 + 11.5i)T^{2} \)
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 \( 1 + (0.174 + 0.649i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.51 - 2.28i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 9.75T + 47T^{2} \)
53 \( 1 + (-3.16 - 3.16i)T + 53iT^{2} \)
59 \( 1 + (11.7 + 3.14i)T + (51.0 + 29.5i)T^{2} \)
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 \( 1 + (-4.46 - 1.19i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 + 1.59iT - 79T^{2} \)
83 \( 1 + 7.57T + 83T^{2} \)
89 \( 1 + (1.21 + 4.54i)T + (-77.0 + 44.5i)T^{2} \)
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

−9.886259563878319325645835204975, −9.063581127621240230940540996530, −8.214962736404883627961236136390, −7.38069128185452210406845807205, −6.73550577767773583389486207404, −5.73966325266268790683145334903, −4.31260568747681400707897549377, −3.95448044125985351484206889874, −3.06571671996911120790685882717, −1.04703815158443639442378688784, 2.30260352543598557120806378710, 2.95849149311839495224620385358, 3.69752546918475129068920871830, 4.46114918038890962847188240163, 5.77300806382504610724631189256, 6.69556194470941698557382213678, 7.85013132681904359255727915539, 8.841253200348573528534384412662, 9.255340441487114465941450446554, 10.42780698842025180818387344678

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