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Properties

Label 2-735-35.13-c1-0-25
Degree $2$
Conductor $735$
Sign $0.948 + 0.316i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
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  + (−1.84 + 1.84i)2-s + (0.707 − 0.707i)3-s − 4.82i·4-s + (0.864 − 2.06i)5-s + 2.61i·6-s + (5.21 + 5.21i)8-s − 1.00i·9-s + (2.21 + 5.40i)10-s + 4.66·11-s + (−3.41 − 3.41i)12-s + (1.58 − 1.58i)13-s + (−0.846 − 2.06i)15-s − 9.61·16-s + (−0.610 − 0.610i)17-s + (1.84 + 1.84i)18-s − 6.51·19-s + ⋯
L(s)  = 1  + (−1.30 + 1.30i)2-s + (0.408 − 0.408i)3-s − 2.41i·4-s + (0.386 − 0.922i)5-s + 1.06i·6-s + (1.84 + 1.84i)8-s − 0.333i·9-s + (0.699 + 1.70i)10-s + 1.40·11-s + (−0.984 − 0.984i)12-s + (0.440 − 0.440i)13-s + (−0.218 − 0.534i)15-s − 2.40·16-s + (−0.148 − 0.148i)17-s + (0.435 + 0.435i)18-s − 1.49·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.948 + 0.316i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.958063 - 0.155864i\)
\(L(\frac12)\) \(\approx\) \(0.958063 - 0.155864i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.864 + 2.06i)T \)
7 \( 1 \)
good2 \( 1 + (1.84 - 1.84i)T - 2iT^{2} \)
11 \( 1 - 4.66T + 11T^{2} \)
13 \( 1 + (-1.58 + 1.58i)T - 13iT^{2} \)
17 \( 1 + (0.610 + 0.610i)T + 17iT^{2} \)
19 \( 1 + 6.51T + 19T^{2} \)
23 \( 1 + (-6.25 - 6.25i)T + 23iT^{2} \)
29 \( 1 + 4.27iT - 29T^{2} \)
31 \( 1 + 3.96iT - 31T^{2} \)
37 \( 1 + (0.286 - 0.286i)T - 37iT^{2} \)
41 \( 1 + 10.7iT - 41T^{2} \)
43 \( 1 + (2.00 + 2.00i)T + 43iT^{2} \)
47 \( 1 + (-3.06 - 3.06i)T + 47iT^{2} \)
53 \( 1 + (-2.32 - 2.32i)T + 53iT^{2} \)
59 \( 1 - 6.70T + 59T^{2} \)
61 \( 1 + 0.703iT - 61T^{2} \)
67 \( 1 + (-6.63 + 6.63i)T - 67iT^{2} \)
71 \( 1 + 9.30T + 71T^{2} \)
73 \( 1 + (4.04 - 4.04i)T - 73iT^{2} \)
79 \( 1 + 0.133iT - 79T^{2} \)
83 \( 1 + (-1.51 + 1.51i)T - 83iT^{2} \)
89 \( 1 + 1.40T + 89T^{2} \)
97 \( 1 + (-3.21 - 3.21i)T + 97iT^{2} \)
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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

−9.810326283735667456167637353474, −9.034299652776953268365621978668, −8.771868406150722078654630120455, −7.87263438973718630867940176823, −6.96686633473510593637896492530, −6.20248969949820546257724841993, −5.42640790508623911808040629264, −4.08845660772232634142884207288, −1.87736816643024421263902932993, −0.817385148503602498792717186396, 1.47783902231620766653357219892, 2.55433003317044837206110110681, 3.48397149009364133718360803954, 4.38866978871859299580518017576, 6.46358568697806951127189889833, 7.06339732752411774187023004152, 8.503773141060049860226138848207, 8.816601887370053680320568440148, 9.664290429078763044557413830557, 10.44181126922262426825021672112

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