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Properties

Label 2-930-15.2-c1-0-43
Degree $2$
Conductor $930$
Sign $-0.245 + 0.969i$
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 + (−0.516 + 1.65i)3-s − 1.00i·4-s + (0.633 + 2.14i)5-s + (0.803 + 1.53i)6-s + (−2.44 − 2.44i)7-s + (−0.707 − 0.707i)8-s + (−2.46 − 1.70i)9-s + (1.96 + 1.06i)10-s − 6.26i·11-s + (1.65 + 0.516i)12-s + (2.06 − 2.06i)13-s − 3.46·14-s + (−3.87 − 0.0614i)15-s − 1.00·16-s + (−5.14 + 5.14i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.298 + 0.954i)3-s − 0.500i·4-s + (0.283 + 0.959i)5-s + (0.327 + 0.626i)6-s + (−0.925 − 0.925i)7-s + (−0.250 − 0.250i)8-s + (−0.821 − 0.569i)9-s + (0.621 + 0.337i)10-s − 1.89i·11-s + (0.477 + 0.149i)12-s + (0.572 − 0.572i)13-s − 0.925·14-s + (−0.999 − 0.0158i)15-s − 0.250·16-s + (−1.24 + 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.245 + 0.969i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.245 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.685027 - 0.879827i\)
\(L(\frac12)\) \(\approx\) \(0.685027 - 0.879827i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.516 - 1.65i)T \)
5 \( 1 + (-0.633 - 2.14i)T \)
31 \( 1 - T \)
good7 \( 1 + (2.44 + 2.44i)T + 7iT^{2} \)
11 \( 1 + 6.26iT - 11T^{2} \)
13 \( 1 + (-2.06 + 2.06i)T - 13iT^{2} \)
17 \( 1 + (5.14 - 5.14i)T - 17iT^{2} \)
19 \( 1 + 3.95iT - 19T^{2} \)
23 \( 1 + (0.977 + 0.977i)T + 23iT^{2} \)
29 \( 1 - 7.01T + 29T^{2} \)
37 \( 1 + (6.92 + 6.92i)T + 37iT^{2} \)
41 \( 1 + 7.39iT - 41T^{2} \)
43 \( 1 + (1.60 - 1.60i)T - 43iT^{2} \)
47 \( 1 + (-7.18 + 7.18i)T - 47iT^{2} \)
53 \( 1 + (-4.50 - 4.50i)T + 53iT^{2} \)
59 \( 1 + 1.19T + 59T^{2} \)
61 \( 1 + 7.46T + 61T^{2} \)
67 \( 1 + (4.48 + 4.48i)T + 67iT^{2} \)
71 \( 1 - 3.67iT - 71T^{2} \)
73 \( 1 + (1.34 - 1.34i)T - 73iT^{2} \)
79 \( 1 - 2.94iT - 79T^{2} \)
83 \( 1 + (-7.44 - 7.44i)T + 83iT^{2} \)
89 \( 1 + 3.81T + 89T^{2} \)
97 \( 1 + (-5.39 - 5.39i)T + 97iT^{2} \)
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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

−10.39658368672707871403502210050, −9.147617807785165089895156667281, −8.469875271144869834726499192299, −6.82624930079949064879990807044, −6.22609277690628381081671381458, −5.54846808781459646245429970938, −4.06745684758598910912740322963, −3.53811521132118039638371855395, −2.72886220681029458535314199289, −0.44264597323760041547031094897, 1.72535253814732381877913419026, 2.72906379796311421231352310964, 4.45001724574812520950641913343, 5.11192796952217539765304720596, 6.19720189978267966604758699878, 6.66363299744030351402181779884, 7.59257818005992753547257868942, 8.629439469382806457278279982468, 9.233914595307796115898794863868, 10.11540836251940341195192983779

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