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Properties

Label 2-1035-15.2-c1-0-11
Degree $2$
Conductor $1035$
Sign $-0.0703 - 0.997i$
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  + (−1.65 + 1.65i)2-s − 3.49i·4-s + (1.35 − 1.77i)5-s + (2.43 + 2.43i)7-s + (2.47 + 2.47i)8-s + (0.691 + 5.19i)10-s − 4.87i·11-s + (−3.02 + 3.02i)13-s − 8.07·14-s − 1.21·16-s + (−4.90 + 4.90i)17-s + 4.20i·19-s + (−6.20 − 4.74i)20-s + (8.07 + 8.07i)22-s + (0.707 + 0.707i)23-s + ⋯
L(s)  = 1  + (−1.17 + 1.17i)2-s − 1.74i·4-s + (0.607 − 0.794i)5-s + (0.921 + 0.921i)7-s + (0.874 + 0.874i)8-s + (0.218 + 1.64i)10-s − 1.46i·11-s + (−0.839 + 0.839i)13-s − 2.15·14-s − 0.302·16-s + (−1.18 + 1.18i)17-s + 0.965i·19-s + (−1.38 − 1.06i)20-s + (1.72 + 1.72i)22-s + (0.147 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0703 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0703 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $-0.0703 - 0.997i$
Analytic conductor: \(8.26451\)
Root analytic conductor: \(2.87480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1035} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1035,\ (\ :1/2),\ -0.0703 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9655495509\)
\(L(\frac12)\) \(\approx\) \(0.9655495509\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.35 + 1.77i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (1.65 - 1.65i)T - 2iT^{2} \)
7 \( 1 + (-2.43 - 2.43i)T + 7iT^{2} \)
11 \( 1 + 4.87iT - 11T^{2} \)
13 \( 1 + (3.02 - 3.02i)T - 13iT^{2} \)
17 \( 1 + (4.90 - 4.90i)T - 17iT^{2} \)
19 \( 1 - 4.20iT - 19T^{2} \)
29 \( 1 - 5.57T + 29T^{2} \)
31 \( 1 - 7.38T + 31T^{2} \)
37 \( 1 + (-3.41 - 3.41i)T + 37iT^{2} \)
41 \( 1 - 6.52iT - 41T^{2} \)
43 \( 1 + (0.560 - 0.560i)T - 43iT^{2} \)
47 \( 1 + (-8.94 + 8.94i)T - 47iT^{2} \)
53 \( 1 + (-9.39 - 9.39i)T + 53iT^{2} \)
59 \( 1 - 1.34T + 59T^{2} \)
61 \( 1 - 6.67T + 61T^{2} \)
67 \( 1 + (-2.51 - 2.51i)T + 67iT^{2} \)
71 \( 1 - 5.54iT - 71T^{2} \)
73 \( 1 + (-11.6 + 11.6i)T - 73iT^{2} \)
79 \( 1 - 6.00iT - 79T^{2} \)
83 \( 1 + (0.475 + 0.475i)T + 83iT^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + (2.46 + 2.46i)T + 97iT^{2} \)
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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

−9.806915691022566698912097427479, −8.952567527676872781031991229534, −8.373485974887418641552421513395, −8.185362741673949528397837841583, −6.72347154549935824321879684762, −6.02722463017669711945278616568, −5.39914890804480698630774945210, −4.36761487658902537444128506912, −2.31035424278382161895209618292, −1.13036182358244070953326221272, 0.77681864560765907684677216042, 2.27128770739864427561535913248, 2.64909454620221584015357009300, 4.26356636004780544223108990461, 5.11018353942962931869605806929, 6.91709647732336426516978832025, 7.27775785375548281691390504585, 8.186817629277924647333101318126, 9.273252038186737094630383835711, 9.859472520760046376692292851704

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