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Properties

Label 2-285-15.2-c1-0-10
Degree $2$
Conductor $285$
Sign $0.434 - 0.900i$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
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  + (0.637 − 0.637i)2-s + (0.0191 + 1.73i)3-s + 1.18i·4-s + (−1.18 − 1.89i)5-s + (1.11 + 1.09i)6-s + (2.03 + 2.03i)7-s + (2.03 + 2.03i)8-s + (−2.99 + 0.0661i)9-s + (−1.96 − 0.458i)10-s + 2.29i·11-s + (−2.05 + 0.0226i)12-s + (−2.17 + 2.17i)13-s + 2.58·14-s + (3.26 − 2.08i)15-s + 0.219·16-s + (4.12 − 4.12i)17-s + ⋯
L(s)  = 1  + (0.450 − 0.450i)2-s + (0.0110 + 0.999i)3-s + 0.593i·4-s + (−0.527 − 0.849i)5-s + (0.455 + 0.445i)6-s + (0.767 + 0.767i)7-s + (0.718 + 0.718i)8-s + (−0.999 + 0.0220i)9-s + (−0.621 − 0.144i)10-s + 0.691i·11-s + (−0.593 + 0.00654i)12-s + (−0.603 + 0.603i)13-s + 0.692·14-s + (0.843 − 0.537i)15-s + 0.0548·16-s + (1.00 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.434 - 0.900i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 0.434 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30173 + 0.816869i\)
\(L(\frac12)\) \(\approx\) \(1.30173 + 0.816869i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0191 - 1.73i)T \)
5 \( 1 + (1.18 + 1.89i)T \)
19 \( 1 - iT \)
good2 \( 1 + (-0.637 + 0.637i)T - 2iT^{2} \)
7 \( 1 + (-2.03 - 2.03i)T + 7iT^{2} \)
11 \( 1 - 2.29iT - 11T^{2} \)
13 \( 1 + (2.17 - 2.17i)T - 13iT^{2} \)
17 \( 1 + (-4.12 + 4.12i)T - 17iT^{2} \)
23 \( 1 + (-0.124 - 0.124i)T + 23iT^{2} \)
29 \( 1 - 3.99T + 29T^{2} \)
31 \( 1 - 8.55T + 31T^{2} \)
37 \( 1 + (7.98 + 7.98i)T + 37iT^{2} \)
41 \( 1 + 4.02iT - 41T^{2} \)
43 \( 1 + (-5.82 + 5.82i)T - 43iT^{2} \)
47 \( 1 + (-2.25 + 2.25i)T - 47iT^{2} \)
53 \( 1 + (6.79 + 6.79i)T + 53iT^{2} \)
59 \( 1 + 0.588T + 59T^{2} \)
61 \( 1 - 4.76T + 61T^{2} \)
67 \( 1 + (-3.10 - 3.10i)T + 67iT^{2} \)
71 \( 1 + 4.72iT - 71T^{2} \)
73 \( 1 + (2.06 - 2.06i)T - 73iT^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 + (10.2 + 10.2i)T + 83iT^{2} \)
89 \( 1 - 0.931T + 89T^{2} \)
97 \( 1 + (-12.3 - 12.3i)T + 97iT^{2} \)
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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

−11.99675270162559295676426648219, −11.44453441799031777371355957238, −10.16181685153481665100328447433, −9.089079235726343549350080608829, −8.361259906489999395461122880995, −7.40089648268414883114115997121, −5.26964114335551466173494012035, −4.78783845710181322422726587692, −3.77870518258543542865538121559, −2.35984705838928438575971770395, 1.14111160808372373566375111095, 3.06450922479804861375247762471, 4.59493096847164330161964547886, 5.88820550128842789675536085674, 6.69138746062092153567390941719, 7.65955413013885003096416056602, 8.230010798984399004590547889872, 10.11994142480675783016665600200, 10.77215340082622986883089188551, 11.67531437900767273887971194818

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