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Properties

Label 2-735-105.104-c1-0-14
Degree $2$
Conductor $735$
Sign $0.975 - 0.221i$
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  − 1.51·2-s + (−1.66 − 0.476i)3-s + 0.294·4-s + (0.775 − 2.09i)5-s + (2.52 + 0.722i)6-s + 2.58·8-s + (2.54 + 1.58i)9-s + (−1.17 + 3.17i)10-s + 2.14i·11-s + (−0.490 − 0.140i)12-s − 3.48·13-s + (−2.29 + 3.12i)15-s − 4.50·16-s + 3.57i·17-s + (−3.85 − 2.40i)18-s − 1.22i·19-s + ⋯
L(s)  = 1  − 1.07·2-s + (−0.961 − 0.275i)3-s + 0.147·4-s + (0.346 − 0.937i)5-s + (1.02 + 0.294i)6-s + 0.913·8-s + (0.848 + 0.529i)9-s + (−0.371 + 1.00i)10-s + 0.647i·11-s + (−0.141 − 0.0405i)12-s − 0.965·13-s + (−0.591 + 0.806i)15-s − 1.12·16-s + 0.867i·17-s + (−0.908 − 0.566i)18-s − 0.280i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.975 - 0.221i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.975 - 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.500436 + 0.0562466i\)
\(L(\frac12)\) \(\approx\) \(0.500436 + 0.0562466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.66 + 0.476i)T \)
5 \( 1 + (-0.775 + 2.09i)T \)
7 \( 1 \)
good2 \( 1 + 1.51T + 2T^{2} \)
11 \( 1 - 2.14iT - 11T^{2} \)
13 \( 1 + 3.48T + 13T^{2} \)
17 \( 1 - 3.57iT - 17T^{2} \)
19 \( 1 + 1.22iT - 19T^{2} \)
23 \( 1 - 1.51T + 23T^{2} \)
29 \( 1 - 5.95iT - 29T^{2} \)
31 \( 1 + 3.17iT - 31T^{2} \)
37 \( 1 - 7.80iT - 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 2.99iT - 43T^{2} \)
47 \( 1 - 6.10iT - 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 2.16T + 59T^{2} \)
61 \( 1 + 3.39iT - 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 - 6.85T + 73T^{2} \)
79 \( 1 + 1.88T + 79T^{2} \)
83 \( 1 + 9.10iT - 83T^{2} \)
89 \( 1 - 1.77T + 89T^{2} \)
97 \( 1 - 1.32T + 97T^{2} \)
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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

−10.20195822778352397853823661247, −9.630888394188933091757836218470, −8.826538605411777022506774960746, −7.83986423520135101495408966361, −7.16334546629806346446861757023, −6.03218304013966704791314865584, −4.94303605644368709404316992082, −4.40253254363522616379739570003, −2.02674578096365277978578099842, −0.934157258717622031828074566707, 0.58413804265747357271270940679, 2.33943488734371595700592440606, 3.90642701935412226777154981957, 5.09101692645056914478478046596, 6.00054361514303294328229248228, 7.09619458023411752185073348117, 7.57143344896631631514888219502, 8.918102659414848953927303949689, 9.686775526056532084350634735722, 10.23542909689979247025013858767

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