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Properties

Label 2-2835-315.293-c0-0-1
Degree $2$
Conductor $2835$
Sign $0.801 + 0.597i$
Analytic cond. $1.41484$
Root an. cond. $1.18947$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)2-s + (0.965 − 0.258i)5-s + (0.5 + 0.866i)7-s + (0.707 − 0.707i)8-s − i·10-s + (−1.22 + 0.707i)11-s + (0.366 + 1.36i)13-s + (0.965 − 0.258i)14-s + (−0.5 − 0.866i)16-s + (−0.707 + 0.707i)17-s + 19-s + (0.366 + 1.36i)22-s + (−0.258 − 0.965i)23-s + (0.866 − 0.499i)25-s + 1.41·26-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (0.965 − 0.258i)5-s + (0.5 + 0.866i)7-s + (0.707 − 0.707i)8-s − i·10-s + (−1.22 + 0.707i)11-s + (0.366 + 1.36i)13-s + (0.965 − 0.258i)14-s + (−0.5 − 0.866i)16-s + (−0.707 + 0.707i)17-s + 19-s + (0.366 + 1.36i)22-s + (−0.258 − 0.965i)23-s + (0.866 − 0.499i)25-s + 1.41·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2835\)    =    \(3^{4} \cdot 5 \cdot 7\)
Sign: $0.801 + 0.597i$
Analytic conductor: \(1.41484\)
Root analytic conductor: \(1.18947\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2835} (2078, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2835,\ (\ :0),\ 0.801 + 0.597i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.929100186\)
\(L(\frac12)\) \(\approx\) \(1.929100186\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.091842448431212075022971356729, −8.310967415000583870660116940363, −7.36638860826879873365893506291, −6.54113084286608228679185775503, −5.69052233363673209598990398611, −4.83843405493955903263452960499, −4.20519399287599994929780440874, −2.89268367864770540438659283479, −2.10432090071041048414089705210, −1.69067916094587954425347107375, 1.25560904933219782135934104531, 2.50814225930910585847458600436, 3.39986208141036299490156512662, 4.79588706888379948779018938300, 5.48474827930793682601758413556, 5.76041998054564081019965803431, 6.84216997709932410841533116849, 7.49513336554350351685245673434, 7.950406214537544147540246496202, 8.851214569993272235900063209583

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