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Properties

Label 2-2535-195.134-c0-0-4
Degree $2$
Conductor $2535$
Sign $-0.993 - 0.114i$
Analytic cond. $1.26512$
Root an. cond. $1.12477$
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  + (1.56 + 0.900i)2-s + (−0.5 + 0.866i)3-s + (1.12 + 1.94i)4-s + i·5-s + (−1.56 + 0.900i)6-s + 2.24i·8-s + (−0.499 − 0.866i)9-s + (−0.900 + 1.56i)10-s − 2.24·12-s + (−0.866 − 0.5i)15-s + (−0.900 + 1.56i)16-s + (−0.222 − 0.385i)17-s − 1.80i·18-s + (1.07 − 0.623i)19-s + (−1.94 + 1.12i)20-s + ⋯
L(s)  = 1  + (1.56 + 0.900i)2-s + (−0.5 + 0.866i)3-s + (1.12 + 1.94i)4-s + i·5-s + (−1.56 + 0.900i)6-s + 2.24i·8-s + (−0.499 − 0.866i)9-s + (−0.900 + 1.56i)10-s − 2.24·12-s + (−0.866 − 0.5i)15-s + (−0.900 + 1.56i)16-s + (−0.222 − 0.385i)17-s − 1.80i·18-s + (1.07 − 0.623i)19-s + (−1.94 + 1.12i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.993 - 0.114i$
Analytic conductor: \(1.26512\)
Root analytic conductor: \(1.12477\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :0),\ -0.993 - 0.114i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.391153455\)
\(L(\frac12)\) \(\approx\) \(2.391153455\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 + (-1.56 - 0.900i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.07 + 0.623i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 0.445iT - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + 1.24iT - T^{2} \)
53 \( 1 - 1.24T + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 + 0.445iT - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.600338759107137951567890088882, −8.495206113049819747472578879530, −7.34272207822646866869528795055, −7.08674417513690616724669549957, −6.05181622836961634844022854696, −5.60366771371839061790482118323, −4.87405181297169151891687907485, −3.88345272840597338021136427840, −3.43608947066968264634690559417, −2.48888404102062232167408787632, 1.06098987890519883167740662159, 1.93443594555486876562687390799, 2.87645170380012660845410553214, 4.08312208314771672193361755612, 4.68691476386744147079380427509, 5.57399493147614133431160078117, 5.96514677031075678176715890757, 6.85221549343554685202455779173, 7.87757575311230118120612171469, 8.643632257675272343834042831187

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