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Properties

Label 2-735-35.13-c1-0-22
Degree $2$
Conductor $735$
Sign $0.803 + 0.595i$
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  + (0.709 − 0.709i)2-s + (0.707 − 0.707i)3-s + 0.992i·4-s + (−2.20 − 0.347i)5-s − 1.00i·6-s + (2.12 + 2.12i)8-s − 1.00i·9-s + (−1.81 + 1.32i)10-s + 3.56·11-s + (0.701 + 0.701i)12-s + (2.78 − 2.78i)13-s + (−1.80 + 1.31i)15-s + 1.02·16-s + (0.370 + 0.370i)17-s + (−0.709 − 0.709i)18-s + 4.12·19-s + ⋯
L(s)  = 1  + (0.501 − 0.501i)2-s + (0.408 − 0.408i)3-s + 0.496i·4-s + (−0.987 − 0.155i)5-s − 0.409i·6-s + (0.750 + 0.750i)8-s − 0.333i·9-s + (−0.573 + 0.417i)10-s + 1.07·11-s + (0.202 + 0.202i)12-s + (0.772 − 0.772i)13-s + (−0.466 + 0.339i)15-s + 0.257·16-s + (0.0899 + 0.0899i)17-s + (−0.167 − 0.167i)18-s + 0.946·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.803 + 0.595i$
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: \((2,\ 735,\ (\ :1/2),\ 0.803 + 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07516 - 0.685123i\)
\(L(\frac12)\) \(\approx\) \(2.07516 - 0.685123i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 + 0.347i)T \)
7 \( 1 \)
good2 \( 1 + (-0.709 + 0.709i)T - 2iT^{2} \)
11 \( 1 - 3.56T + 11T^{2} \)
13 \( 1 + (-2.78 + 2.78i)T - 13iT^{2} \)
17 \( 1 + (-0.370 - 0.370i)T + 17iT^{2} \)
19 \( 1 - 4.12T + 19T^{2} \)
23 \( 1 + (-1.82 - 1.82i)T + 23iT^{2} \)
29 \( 1 + 6.14iT - 29T^{2} \)
31 \( 1 - 1.97iT - 31T^{2} \)
37 \( 1 + (0.111 - 0.111i)T - 37iT^{2} \)
41 \( 1 + 8.28iT - 41T^{2} \)
43 \( 1 + (-9.01 - 9.01i)T + 43iT^{2} \)
47 \( 1 + (3.80 + 3.80i)T + 47iT^{2} \)
53 \( 1 + (4.06 + 4.06i)T + 53iT^{2} \)
59 \( 1 + 2.60T + 59T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 + (3.89 - 3.89i)T - 67iT^{2} \)
71 \( 1 + 7.23T + 71T^{2} \)
73 \( 1 + (10.8 - 10.8i)T - 73iT^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 + (9.42 - 9.42i)T - 83iT^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + (-2.48 - 2.48i)T + 97iT^{2} \)
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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

−10.56668481930876345369158900933, −9.242919427877409251897185735868, −8.463800899343169024173538504381, −7.74164803265092272843045065442, −7.05737859609135364446933787186, −5.74365185696498185742605874730, −4.41781052457281586454723816045, −3.65372962812032076838348426949, −2.90936120551561261671011573585, −1.25505469435135422736262395406, 1.31941664051521241839396175324, 3.27702932617250667834310897957, 4.13416302490530619091744891794, 4.85711866282997807293596746715, 6.10660852673193478297596510993, 6.91151427659970513539565931484, 7.69879810298719893653376309782, 8.880869281570838900184354435036, 9.421849004240065988693127631591, 10.57266732002559206916497113768

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