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Properties

Label 2-735-105.53-c1-0-26
Degree $2$
Conductor $735$
Sign $0.123 + 0.992i$
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  + (0.565 − 2.10i)2-s + (−0.871 + 1.49i)3-s + (−2.39 − 1.38i)4-s + (−2.05 + 0.892i)5-s + (2.66 + 2.68i)6-s + (−1.18 + 1.18i)8-s + (−1.48 − 2.60i)9-s + (0.723 + 4.82i)10-s + (2.93 + 1.69i)11-s + (4.16 − 2.38i)12-s + (0.206 + 0.206i)13-s + (0.451 − 3.84i)15-s + (−0.935 − 1.62i)16-s + (0.228 − 0.0612i)17-s + (−6.34 + 1.65i)18-s + (4.60 − 2.65i)19-s + ⋯
L(s)  = 1  + (0.399 − 1.49i)2-s + (−0.503 + 0.864i)3-s + (−1.19 − 0.692i)4-s + (−0.916 + 0.399i)5-s + (1.08 + 1.09i)6-s + (−0.419 + 0.419i)8-s + (−0.493 − 0.869i)9-s + (0.228 + 1.52i)10-s + (0.883 + 0.510i)11-s + (1.20 − 0.687i)12-s + (0.0573 + 0.0573i)13-s + (0.116 − 0.993i)15-s + (−0.233 − 0.405i)16-s + (0.0554 − 0.0148i)17-s + (−1.49 + 0.388i)18-s + (1.05 − 0.609i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.123 + 0.992i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.123 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01076 - 0.893174i\)
\(L(\frac12)\) \(\approx\) \(1.01076 - 0.893174i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.871 - 1.49i)T \)
5 \( 1 + (2.05 - 0.892i)T \)
7 \( 1 \)
good2 \( 1 + (-0.565 + 2.10i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-2.93 - 1.69i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.206 - 0.206i)T + 13iT^{2} \)
17 \( 1 + (-0.228 + 0.0612i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.60 + 2.65i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.93 - 1.85i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.84T + 29T^{2} \)
31 \( 1 + (-4.55 + 7.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.19 - 1.92i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.0314iT - 41T^{2} \)
43 \( 1 + (3.76 + 3.76i)T + 43iT^{2} \)
47 \( 1 + (-1.30 + 4.87i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.30 - 4.85i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.15 - 8.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.40 + 5.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.32 - 8.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.95iT - 71T^{2} \)
73 \( 1 + (-11.7 + 3.15i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.91 + 5.72i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.88 - 3.88i)T - 83iT^{2} \)
89 \( 1 + (-1.00 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.26 - 2.26i)T - 97iT^{2} \)
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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

−10.45917336775239297518079250240, −9.571898268598435722443257325536, −9.065655713492317329211023041990, −7.55935416227831741014767770989, −6.57562829762928790039360800322, −5.16181310768776819954825903004, −4.34706933969156985442518587068, −3.63940725789017452660082290712, −2.76088429247463177238643966258, −0.882633930377268273940781735384, 1.10798718726417979982605370624, 3.34633202096979061509230481497, 4.63619624237007971856643205500, 5.36481010316247648764560867847, 6.34958790490858349707112883177, 6.98723665386575291304679583582, 7.79000616869931991330035056401, 8.378800473190411548835979048026, 9.254533272796924285696697410037, 10.91055268465941351424336251999

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