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Properties

Label 2-2070-15.2-c1-0-4
Degree $2$
Conductor $2070$
Sign $-0.896 - 0.442i$
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 + (0.125 − 2.23i)5-s + (3.50 + 3.50i)7-s + (0.707 + 0.707i)8-s + (1.48 + 1.66i)10-s + 0.764i·11-s + (−3.69 + 3.69i)13-s − 4.96·14-s − 1.00·16-s + (−0.764 + 0.764i)17-s + 6.41i·19-s + (−2.23 − 0.125i)20-s + (−0.540 − 0.540i)22-s + (−0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.0562 − 0.998i)5-s + (1.32 + 1.32i)7-s + (0.250 + 0.250i)8-s + (0.471 + 0.527i)10-s + 0.230i·11-s + (−1.02 + 1.02i)13-s − 1.32·14-s − 0.250·16-s + (−0.185 + 0.185i)17-s + 1.47i·19-s + (−0.499 − 0.0281i)20-s + (−0.115 − 0.115i)22-s + (−0.147 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.896 - 0.442i$
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: \((2,\ 2070,\ (\ :1/2),\ -0.896 - 0.442i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8027938564\)
\(L(\frac12)\) \(\approx\) \(0.8027938564\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.125 + 2.23i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-3.50 - 3.50i)T + 7iT^{2} \)
11 \( 1 - 0.764iT - 11T^{2} \)
13 \( 1 + (3.69 - 3.69i)T - 13iT^{2} \)
17 \( 1 + (0.764 - 0.764i)T - 17iT^{2} \)
19 \( 1 - 6.41iT - 19T^{2} \)
29 \( 1 + 1.67T + 29T^{2} \)
31 \( 1 + 9.93T + 31T^{2} \)
37 \( 1 + (8.31 + 8.31i)T + 37iT^{2} \)
41 \( 1 - 10.5iT - 41T^{2} \)
43 \( 1 + (-2.96 + 2.96i)T - 43iT^{2} \)
47 \( 1 + (-2.54 + 2.54i)T - 47iT^{2} \)
53 \( 1 + (9.13 + 9.13i)T + 53iT^{2} \)
59 \( 1 + 0.219T + 59T^{2} \)
61 \( 1 + 1.82T + 61T^{2} \)
67 \( 1 + (-5.38 - 5.38i)T + 67iT^{2} \)
71 \( 1 + 6.14iT - 71T^{2} \)
73 \( 1 + (1.36 - 1.36i)T - 73iT^{2} \)
79 \( 1 + 4.66iT - 79T^{2} \)
83 \( 1 + (1.48 + 1.48i)T + 83iT^{2} \)
89 \( 1 - 1.81T + 89T^{2} \)
97 \( 1 + (-8.54 - 8.54i)T + 97iT^{2} \)
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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

−9.171714317212196033894922701696, −8.744658697226006678313820339447, −7.971417856477700720469905783644, −7.41216090667919129002003168431, −6.19146665317132161623668323075, −5.37253026464439180382596210429, −4.96117479113132824849354835319, −3.97945771280391781175212280893, −2.01547868273806380099204358969, −1.74803628289156630394981000674, 0.31647922661653567074887062291, 1.69873055586618181215135858445, 2.74258736259354691286813325896, 3.64939400370985760398983411422, 4.63549355635914945022250041054, 5.44259581624360645996564832989, 6.87472638190024663702404207964, 7.37764930581388837628121790849, 7.82643850047071575315611910343, 8.820936460890785536730382975233

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