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Properties

Label 2-1035-15.8-c1-0-1
Degree $2$
Conductor $1035$
Sign $-0.720 - 0.693i$
Analytic cond. $8.26451$
Root an. cond. $2.87480$
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  + (−1.47 − 1.47i)2-s + 2.34i·4-s + (−0.834 + 2.07i)5-s + (0.188 − 0.188i)7-s + (0.502 − 0.502i)8-s + (4.28 − 1.82i)10-s + 2.92i·11-s + (0.0612 + 0.0612i)13-s − 0.556·14-s + 3.20·16-s + (−1.41 − 1.41i)17-s − 1.33i·19-s + (−4.85 − 1.95i)20-s + (4.30 − 4.30i)22-s + (−0.707 + 0.707i)23-s + ⋯
L(s)  = 1  + (−1.04 − 1.04i)2-s + 1.17i·4-s + (−0.373 + 0.927i)5-s + (0.0713 − 0.0713i)7-s + (0.177 − 0.177i)8-s + (1.35 − 0.577i)10-s + 0.880i·11-s + (0.0169 + 0.0169i)13-s − 0.148·14-s + 0.800·16-s + (−0.342 − 0.342i)17-s − 0.306i·19-s + (−1.08 − 0.436i)20-s + (0.917 − 0.917i)22-s + (−0.147 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $-0.720 - 0.693i$
Analytic conductor: \(8.26451\)
Root analytic conductor: \(2.87480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1035} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1035,\ (\ :1/2),\ -0.720 - 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07162534572\)
\(L(\frac12)\) \(\approx\) \(0.07162534572\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.834 - 2.07i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.47 + 1.47i)T + 2iT^{2} \)
7 \( 1 + (-0.188 + 0.188i)T - 7iT^{2} \)
11 \( 1 - 2.92iT - 11T^{2} \)
13 \( 1 + (-0.0612 - 0.0612i)T + 13iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 + 1.33iT - 19T^{2} \)
29 \( 1 + 4.81T + 29T^{2} \)
31 \( 1 - 4.71T + 31T^{2} \)
37 \( 1 + (1.78 - 1.78i)T - 37iT^{2} \)
41 \( 1 - 2.01iT - 41T^{2} \)
43 \( 1 + (5.51 + 5.51i)T + 43iT^{2} \)
47 \( 1 + (7.37 + 7.37i)T + 47iT^{2} \)
53 \( 1 + (-3.62 + 3.62i)T - 53iT^{2} \)
59 \( 1 + 5.14T + 59T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 + (4.95 - 4.95i)T - 67iT^{2} \)
71 \( 1 + 7.80iT - 71T^{2} \)
73 \( 1 + (8.46 + 8.46i)T + 73iT^{2} \)
79 \( 1 - 5.81iT - 79T^{2} \)
83 \( 1 + (0.630 - 0.630i)T - 83iT^{2} \)
89 \( 1 - 2.89T + 89T^{2} \)
97 \( 1 + (1.69 - 1.69i)T - 97iT^{2} \)
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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

−10.27554926862330067063661691955, −9.628832265518275165743059802564, −8.812943398038332948934623622509, −7.86412160861746511688274171192, −7.20319652910578932113862932864, −6.22723468196555388047337959898, −4.83276379201636500885027887067, −3.62865711636024668098086541026, −2.69638903330051348789871805522, −1.72718990070479748997234247202, 0.04756316504459599684442081029, 1.41185569972596004065818745645, 3.33960555528583599146981795318, 4.53022592641031079734279013829, 5.64703395582713539018792912342, 6.27998832292935970764493114617, 7.36413319854034496517084184423, 8.114422201382705292511149143959, 8.616294601979255768213307403583, 9.282244093680499461043199780431

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