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Properties

Label 2-845-65.18-c1-0-52
Degree $2$
Conductor $845$
Sign $0.999 - 0.0153i$
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  − 0.274i·2-s + (1.67 + 1.67i)3-s + 1.92·4-s + (1.45 − 1.69i)5-s + (0.459 − 0.459i)6-s + 0.386·7-s − 1.07i·8-s + 2.58i·9-s + (−0.466 − 0.399i)10-s + (−3.08 − 3.08i)11-s + (3.21 + 3.21i)12-s − 0.106i·14-s + (5.26 − 0.409i)15-s + 3.55·16-s + (−1.39 − 1.39i)17-s + 0.710·18-s + ⋯
L(s)  = 1  − 0.194i·2-s + (0.964 + 0.964i)3-s + 0.962·4-s + (0.650 − 0.759i)5-s + (0.187 − 0.187i)6-s + 0.145·7-s − 0.381i·8-s + 0.861i·9-s + (−0.147 − 0.126i)10-s + (−0.929 − 0.929i)11-s + (0.928 + 0.928i)12-s − 0.0283i·14-s + (1.36 − 0.105i)15-s + 0.888·16-s + (−0.338 − 0.338i)17-s + 0.167·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.999 - 0.0153i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (408, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.999 - 0.0153i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.88989 + 0.0221645i\)
\(L(\frac12)\) \(\approx\) \(2.88989 + 0.0221645i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.45 + 1.69i)T \)
13 \( 1 \)
good2 \( 1 + 0.274iT - 2T^{2} \)
3 \( 1 + (-1.67 - 1.67i)T + 3iT^{2} \)
7 \( 1 - 0.386T + 7T^{2} \)
11 \( 1 + (3.08 + 3.08i)T + 11iT^{2} \)
17 \( 1 + (1.39 + 1.39i)T + 17iT^{2} \)
19 \( 1 + (-3.54 - 3.54i)T + 19iT^{2} \)
23 \( 1 + (0.235 - 0.235i)T - 23iT^{2} \)
29 \( 1 - 8.16iT - 29T^{2} \)
31 \( 1 + (2.54 - 2.54i)T - 31iT^{2} \)
37 \( 1 - 4.82T + 37T^{2} \)
41 \( 1 + (3.29 - 3.29i)T - 41iT^{2} \)
43 \( 1 + (4.82 - 4.82i)T - 43iT^{2} \)
47 \( 1 + 9.83T + 47T^{2} \)
53 \( 1 + (7.17 + 7.17i)T + 53iT^{2} \)
59 \( 1 + (1.71 - 1.71i)T - 59iT^{2} \)
61 \( 1 - 10.6T + 61T^{2} \)
67 \( 1 - 6.37iT - 67T^{2} \)
71 \( 1 + (3.07 - 3.07i)T - 71iT^{2} \)
73 \( 1 - 6.08iT - 73T^{2} \)
79 \( 1 - 3.34iT - 79T^{2} \)
83 \( 1 - 5.18T + 83T^{2} \)
89 \( 1 + (3.53 - 3.53i)T - 89iT^{2} \)
97 \( 1 + 14.7iT - 97T^{2} \)
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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

−9.994063071058615263260426453591, −9.577333865967520718191297516225, −8.453712234126871896950602116560, −8.053752333057241209463219276928, −6.73028041454928267556159711232, −5.61614223267892496817075050067, −4.85921910199126947393599984166, −3.46226279255571799327655221384, −2.82585707681767409687974846974, −1.52816204998040045852550528471, 1.82649452496407735372914830778, 2.38344948395518168897623577202, 3.23446939484160125320823040752, 5.04117589408957876972175099942, 6.19851799440128426012392063050, 6.90538788343341516028088137002, 7.60757875466253561027510628860, 8.077429191262691468732138574962, 9.372440719181474994592560724719, 10.15121760370021609588218053593

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