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Properties

Label 2-2070-15.2-c1-0-18
Degree $2$
Conductor $2070$
Sign $0.927 + 0.374i$
Analytic cond. $16.5290$
Root an. cond. $4.06559$
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 − 1.00i·4-s + (−2.12 − 0.707i)5-s + (2 + 2i)7-s + (−0.707 − 0.707i)8-s + (−2 + 0.999i)10-s + 1.41i·11-s + (2 − 2i)13-s + 2.82·14-s − 1.00·16-s + (−1.41 + 1.41i)17-s + 2i·19-s + (−0.707 + 2.12i)20-s + (1.00 + 1.00i)22-s + (0.707 + 0.707i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.948 − 0.316i)5-s + (0.755 + 0.755i)7-s + (−0.250 − 0.250i)8-s + (−0.632 + 0.316i)10-s + 0.426i·11-s + (0.554 − 0.554i)13-s + 0.755·14-s − 0.250·16-s + (−0.342 + 0.342i)17-s + 0.458i·19-s + (−0.158 + 0.474i)20-s + (0.213 + 0.213i)22-s + (0.147 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(16.5290\)
Root analytic conductor: \(4.06559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.111304732\)
\(L(\frac12)\) \(\approx\) \(2.111304732\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (2.12 + 0.707i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-2 - 2i)T + 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + (-2 + 2i)T - 13iT^{2} \)
17 \( 1 + (1.41 - 1.41i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + (-3 + 3i)T - 43iT^{2} \)
47 \( 1 + (1.41 - 1.41i)T - 47iT^{2} \)
53 \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (-1 - i)T + 67iT^{2} \)
71 \( 1 + 9.89iT - 71T^{2} \)
73 \( 1 + (-5 + 5i)T - 73iT^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + (-4.24 - 4.24i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−8.944228405361477732692661526068, −8.318931790623904176492883465213, −7.71616702798462470787030056519, −6.60718299238034628297025699430, −5.68621687099995022975155032350, −4.88843785827636482782186136191, −4.20826653558794709000224845624, −3.28398373663606917737454718151, −2.21793595744366559571739786208, −1.02299600809333155323628525897, 0.834955379030803557616010833786, 2.52983610771423755680957152896, 3.63515714409713663846990294394, 4.31519742625111804672541882795, 4.94784783836970070853250990841, 6.14014016941443277597412966644, 6.90754628755663157541191848876, 7.49118827912265553184455921029, 8.291052199463958136482047785040, 8.788135519960628616569897553151

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