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Properties

Label 2-735-105.104-c1-0-56
Degree $2$
Conductor $735$
Sign $-0.709 + 0.704i$
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 + (0.291 − 3.86i)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 + (0.0753 − 0.997i)15-s − 1.12·16-s − 0.867i·17-s + (0.908 − 0.566i)18-s − 0.280i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.709 + 0.704i$
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: \((2,\ 735,\ (\ :1/2),\ -0.709 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0839366 - 0.203718i\)
\(L(\frac12)\) \(\approx\) \(0.0839366 - 0.203718i\)
\(L(\frac{3}{2})\) 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

−10.11603395903263836959425352500, −9.587373124831260259540590268138, −8.184007563182189961248838134310, −6.99382279734711269543810997667, −6.36995251949364551214816259926, −5.42176255155844099094125183464, −4.64452426584128968092730749430, −3.69472151342253410284996595574, −2.69080503484688617720983083181, −0.087111637597271660231347698180, 1.78971060072471725866905761065, 3.61974835770810955237085437236, 4.65120781254780243575294469846, 5.08241545835508513858864412576, 5.98356115136556706648001045116, 6.98056512906889931223861710088, 7.952627070030462721580826684197, 9.043264273616780279083941377979, 9.944311125257716208181376651947, 10.94028142624124357748740050761

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