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Properties

Label 2-465-155.4-c1-0-23
Degree $2$
Conductor $465$
Sign $-0.246 + 0.969i$
Analytic cond. $3.71304$
Root an. cond. $1.92692$
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  + (−1.36 + 0.442i)2-s + (−0.951 − 0.309i)3-s + (0.0421 − 0.0306i)4-s + (2.03 − 0.926i)5-s + 1.43·6-s + (−1.00 − 1.38i)7-s + (1.64 − 2.25i)8-s + (0.809 + 0.587i)9-s + (−2.36 + 2.16i)10-s + (−2.35 + 1.70i)11-s + (−0.0495 + 0.0160i)12-s + (0.880 + 0.286i)13-s + (1.98 + 1.44i)14-s + (−2.22 + 0.251i)15-s + (−1.26 + 3.90i)16-s + (−0.872 + 1.20i)17-s + ⋯
L(s)  = 1  + (−0.963 + 0.313i)2-s + (−0.549 − 0.178i)3-s + (0.0210 − 0.0153i)4-s + (0.910 − 0.414i)5-s + 0.584·6-s + (−0.381 − 0.525i)7-s + (0.579 − 0.798i)8-s + (0.269 + 0.195i)9-s + (−0.747 + 0.683i)10-s + (−0.709 + 0.515i)11-s + (−0.0143 + 0.00464i)12-s + (0.244 + 0.0793i)13-s + (0.531 + 0.386i)14-s + (−0.573 + 0.0650i)15-s + (−0.316 + 0.975i)16-s + (−0.211 + 0.291i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.246 + 0.969i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ -0.246 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.263414 - 0.338970i\)
\(L(\frac12)\) \(\approx\) \(0.263414 - 0.338970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (-2.03 + 0.926i)T \)
31 \( 1 + (-3.31 + 4.46i)T \)
good2 \( 1 + (1.36 - 0.442i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + (1.00 + 1.38i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (2.35 - 1.70i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.880 - 0.286i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.872 - 1.20i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.54 + 7.84i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.10 - 5.65i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.36 + 4.19i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 8.97iT - 37T^{2} \)
41 \( 1 + (1.55 + 4.79i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (4.01 - 1.30i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (8.32 + 2.70i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.73 + 6.51i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.267 - 0.823i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 13.2T + 61T^{2} \)
67 \( 1 + 7.44iT - 67T^{2} \)
71 \( 1 + (-10.3 - 7.53i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-9.02 - 12.4i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (11.7 + 8.53i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.277 + 0.0902i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (8.47 - 6.16i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.27 - 1.74i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.43450686236651971354166789198, −9.827206047894572592210502517770, −9.106924557183186393789905623349, −8.088223062185307643063167120111, −7.14281072733141911639654301086, −6.35461576241939819480923683780, −5.17403075321721581439521586980, −4.07975695310853921828389748687, −2.04902376893897127012687223426, −0.38959076096624611236350860947, 1.62443844849849654408115331116, 2.97258582847805207113639384576, 4.78948569724846829081365418391, 5.81448613488683567737753410928, 6.48111964037165205504048735770, 8.012018942636287940614976499224, 8.735964322618701795145581285739, 9.774025682609410557041386047534, 10.32483176289782889245593784273, 10.83419415220745326034891382007

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