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Properties

Label 2-805-35.27-c1-0-86
Degree $2$
Conductor $805$
Sign $0.955 + 0.294i$
Analytic cond. $6.42795$
Root an. cond. $2.53534$
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  + (−1.49 − 1.49i)2-s + (−1.75 − 1.75i)3-s + 2.49i·4-s + (−0.277 − 2.21i)5-s + 5.27i·6-s + (−2.59 − 0.502i)7-s + (0.748 − 0.748i)8-s + 3.18i·9-s + (−2.91 + 3.74i)10-s − 5.46·11-s + (4.39 − 4.39i)12-s + (−2.92 − 2.92i)13-s + (3.14 + 4.64i)14-s + (−3.41 + 4.38i)15-s + 2.75·16-s + (3.96 − 3.96i)17-s + ⋯
L(s)  = 1  + (−1.06 − 1.06i)2-s + (−1.01 − 1.01i)3-s + 1.24i·4-s + (−0.124 − 0.992i)5-s + 2.15i·6-s + (−0.981 − 0.189i)7-s + (0.264 − 0.264i)8-s + 1.06i·9-s + (−0.920 + 1.18i)10-s − 1.64·11-s + (1.26 − 1.26i)12-s + (−0.810 − 0.810i)13-s + (0.839 + 1.24i)14-s + (−0.881 + 1.13i)15-s + 0.688·16-s + (0.961 − 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.955 + 0.294i$
Analytic conductor: \(6.42795\)
Root analytic conductor: \(2.53534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 805,\ (\ :1/2),\ 0.955 + 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0835591 - 0.0125638i\)
\(L(\frac12)\) \(\approx\) \(0.0835591 - 0.0125638i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.277 + 2.21i)T \)
7 \( 1 + (2.59 + 0.502i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (1.49 + 1.49i)T + 2iT^{2} \)
3 \( 1 + (1.75 + 1.75i)T + 3iT^{2} \)
11 \( 1 + 5.46T + 11T^{2} \)
13 \( 1 + (2.92 + 2.92i)T + 13iT^{2} \)
17 \( 1 + (-3.96 + 3.96i)T - 17iT^{2} \)
19 \( 1 + 2.35T + 19T^{2} \)
29 \( 1 - 4.74iT - 29T^{2} \)
31 \( 1 + 1.70iT - 31T^{2} \)
37 \( 1 + (3.00 + 3.00i)T + 37iT^{2} \)
41 \( 1 + 8.57iT - 41T^{2} \)
43 \( 1 + (7.01 - 7.01i)T - 43iT^{2} \)
47 \( 1 + (-5.14 + 5.14i)T - 47iT^{2} \)
53 \( 1 + (-8.99 + 8.99i)T - 53iT^{2} \)
59 \( 1 - 9.30T + 59T^{2} \)
61 \( 1 + 9.11iT - 61T^{2} \)
67 \( 1 + (7.51 + 7.51i)T + 67iT^{2} \)
71 \( 1 + 1.02T + 71T^{2} \)
73 \( 1 + (3.83 + 3.83i)T + 73iT^{2} \)
79 \( 1 + 9.70iT - 79T^{2} \)
83 \( 1 + (-6.88 - 6.88i)T + 83iT^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + (7.56 - 7.56i)T - 97iT^{2} \)
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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

−9.649861531608845518970857821675, −8.582357189358367149110479282705, −7.73540286907855552007926680728, −7.09810493433425735851045357704, −5.61761868774508654611387428628, −5.19202873663822585144440520164, −3.26172585033789310919196884844, −2.15244841581911896465441684362, −0.65734807895753847989187404413, −0.11537461638961459805447812606, 2.77281794124848202502849305196, 4.06305069563370242370019335235, 5.44731790523105834763606878472, 5.98408087056717580825561172693, 6.86111064955083028948166280749, 7.60318533675567895403687905412, 8.583796575915953937639262242088, 9.767613207063515692103401984066, 10.17321777340663884664210695544, 10.51840601050320625897661731207

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