Properties

Label 2-930-155.123-c1-0-7
Degree $2$
Conductor $930$
Sign $-0.158 - 0.987i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (−2.14 − 0.616i)5-s + 1.00i·6-s + (0.0344 − 0.0344i)7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (1.95 − 1.08i)10-s − 0.241i·11-s + (−0.707 − 0.707i)12-s + (−3.88 + 3.88i)13-s + 0.0487i·14-s + (−1.95 + 1.08i)15-s − 1.00·16-s + (−1.35 − 1.35i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (−0.961 − 0.275i)5-s + 0.408i·6-s + (0.0130 − 0.0130i)7-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (0.618 − 0.342i)10-s − 0.0729i·11-s + (−0.204 − 0.204i)12-s + (−1.07 + 1.07i)13-s + 0.0130i·14-s + (−0.504 + 0.279i)15-s − 0.250·16-s + (−0.329 − 0.329i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.158 - 0.987i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.158 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.484754 + 0.568906i\)
\(L(\frac12)\) \(\approx\) \(0.484754 + 0.568906i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (2.14 + 0.616i)T \)
31 \( 1 + (3.04 - 4.66i)T \)
good7 \( 1 + (-0.0344 + 0.0344i)T - 7iT^{2} \)
11 \( 1 + 0.241iT - 11T^{2} \)
13 \( 1 + (3.88 - 3.88i)T - 13iT^{2} \)
17 \( 1 + (1.35 + 1.35i)T + 17iT^{2} \)
19 \( 1 - 8.10iT - 19T^{2} \)
23 \( 1 + (-2.64 + 2.64i)T - 23iT^{2} \)
29 \( 1 - 8.54T + 29T^{2} \)
37 \( 1 + (-4.54 - 4.54i)T + 37iT^{2} \)
41 \( 1 + 1.46T + 41T^{2} \)
43 \( 1 + (-0.329 + 0.329i)T - 43iT^{2} \)
47 \( 1 + (2.81 - 2.81i)T - 47iT^{2} \)
53 \( 1 + (0.597 - 0.597i)T - 53iT^{2} \)
59 \( 1 - 13.6iT - 59T^{2} \)
61 \( 1 + 11.6iT - 61T^{2} \)
67 \( 1 + (0.653 - 0.653i)T - 67iT^{2} \)
71 \( 1 + 4.44T + 71T^{2} \)
73 \( 1 + (8.64 - 8.64i)T - 73iT^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 + (-0.681 + 0.681i)T - 83iT^{2} \)
89 \( 1 - 2.20T + 89T^{2} \)
97 \( 1 + (5.06 - 5.06i)T - 97iT^{2} \)
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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

−10.10215994048070797506160562535, −9.246575175649278544383591394714, −8.479045531844660291082926897115, −7.83333866133883693021735015400, −7.07426112502054269387208784963, −6.34927337211942231545434978633, −4.97616488666044251716153333622, −4.15894787895137873883472068545, −2.80482828892197529144951946955, −1.33525732679281149071827176907, 0.42220656534641499699842763370, 2.49174185175988006520074336852, 3.19027090299804439474128764041, 4.33162728679160326509850601061, 5.13360669986687411892821954719, 6.77885679723579819455906426449, 7.52016668983136187997586124903, 8.240066132531027356116703248579, 9.048639144514382690330056404664, 9.854298439486563563944210934686

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