Properties

Label 2-930-93.26-c1-0-33
Degree $2$
Conductor $930$
Sign $0.790 - 0.612i$
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  + i·2-s + (1.64 + 0.536i)3-s − 4-s + (0.866 − 0.5i)5-s + (−0.536 + 1.64i)6-s + (0.615 − 1.06i)7-s i·8-s + (2.42 + 1.76i)9-s + (0.5 + 0.866i)10-s + (−0.427 − 0.741i)11-s + (−1.64 − 0.536i)12-s + (4.88 − 2.82i)13-s + (1.06 + 0.615i)14-s + (1.69 − 0.358i)15-s + 16-s + (3.42 − 5.93i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.950 + 0.309i)3-s − 0.5·4-s + (0.387 − 0.223i)5-s + (−0.219 + 0.672i)6-s + (0.232 − 0.402i)7-s − 0.353i·8-s + (0.808 + 0.589i)9-s + (0.158 + 0.273i)10-s + (−0.129 − 0.223i)11-s + (−0.475 − 0.154i)12-s + (1.35 − 0.782i)13-s + (0.284 + 0.164i)14-s + (0.437 − 0.0926i)15-s + 0.250·16-s + (0.830 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.790 - 0.612i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.790 - 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28149 + 0.780404i\)
\(L(\frac12)\) \(\approx\) \(2.28149 + 0.780404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.64 - 0.536i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (1.91 - 5.22i)T \)
good7 \( 1 + (-0.615 + 1.06i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.427 + 0.741i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.88 + 2.82i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.42 + 5.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.242 - 0.420i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.95T + 23T^{2} \)
29 \( 1 - 1.17T + 29T^{2} \)
37 \( 1 + (-7.56 - 4.36i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.21 - 3.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.37 + 2.52i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.88iT - 47T^{2} \)
53 \( 1 + (0.346 + 0.599i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.639 + 0.369i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 6.48iT - 61T^{2} \)
67 \( 1 + (-4.07 - 7.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.36 - 1.94i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.44 - 2.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.99 - 1.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.20 - 2.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.45T + 89T^{2} \)
97 \( 1 + 5.55T + 97T^{2} \)
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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

−9.963056948541127307055856917793, −9.236235487982948531440104265895, −8.213209823615197678022031719988, −7.969326407630243922922208916836, −6.85200372531591961752337979257, −5.79892795476533971114184660022, −4.91925266656087003180110672593, −3.86813142363920649896403247308, −2.93321480445221446264826525945, −1.26453467197009871100748753674, 1.55011752278444380609110923654, 2.23327791650494843307552642748, 3.57087223276215164249563786516, 4.16341448814313543430668616554, 5.74219931149978755792460582825, 6.47076755622076825662761866874, 7.80991692078534464421462409143, 8.394601098364879159370989542612, 9.165252197432926353087311268315, 9.971401252762453590024053850298

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