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Properties

Label 2-735-105.2-c1-0-58
Degree $2$
Conductor $735$
Sign $0.726 + 0.687i$
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  + (0.0952 + 0.355i)2-s + (0.904 − 1.47i)3-s + (1.61 − 0.932i)4-s + (2.22 + 0.249i)5-s + (0.611 + 0.180i)6-s + (1.00 + 1.00i)8-s + (−1.36 − 2.67i)9-s + (0.123 + 0.813i)10-s + (−2.93 + 1.69i)11-s + (0.0835 − 3.22i)12-s + (1.59 − 1.59i)13-s + (2.37 − 3.05i)15-s + (1.60 − 2.77i)16-s + (−0.192 − 0.0514i)17-s + (0.820 − 0.739i)18-s + (6.36 + 3.67i)19-s + ⋯
L(s)  = 1  + (0.0673 + 0.251i)2-s + (0.522 − 0.852i)3-s + (0.807 − 0.466i)4-s + (0.993 + 0.111i)5-s + (0.249 + 0.0738i)6-s + (0.355 + 0.355i)8-s + (−0.454 − 0.890i)9-s + (0.0389 + 0.257i)10-s + (−0.884 + 0.510i)11-s + (0.0241 − 0.931i)12-s + (0.442 − 0.442i)13-s + (0.614 − 0.789i)15-s + (0.400 − 0.694i)16-s + (−0.0466 − 0.0124i)17-s + (0.193 − 0.174i)18-s + (1.45 + 0.842i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.726 + 0.687i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40224 - 0.956629i\)
\(L(\frac12)\) \(\approx\) \(2.40224 - 0.956629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.904 + 1.47i)T \)
5 \( 1 + (-2.22 - 0.249i)T \)
7 \( 1 \)
good2 \( 1 + (-0.0952 - 0.355i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (2.93 - 1.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.59 + 1.59i)T - 13iT^{2} \)
17 \( 1 + (0.192 + 0.0514i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-6.36 - 3.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.02 - 0.810i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.49T + 29T^{2} \)
31 \( 1 + (0.461 + 0.798i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.08 - 2.16i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 1.39iT - 41T^{2} \)
43 \( 1 + (-0.864 + 0.864i)T - 43iT^{2} \)
47 \( 1 + (-0.238 - 0.889i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.39 + 8.93i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.12 - 5.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.916 - 1.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.298 - 1.11i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.77iT - 71T^{2} \)
73 \( 1 + (-6.56 - 1.75i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.95 - 1.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.26 + 6.26i)T + 83iT^{2} \)
89 \( 1 + (6.18 - 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.71 + 6.71i)T + 97iT^{2} \)
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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

−10.11496498551266243437621228798, −9.595548939908431784598138669323, −8.343941373282041423533211659953, −7.49271514460136393871159908896, −6.89648602991158854059047435447, −5.74233832508230995722327530220, −5.45535289400777644308532809261, −3.37076369959169044486442258826, −2.28936961266010452847676872941, −1.44890899884645678329468492121, 1.89796422229224059110885986474, 2.86312122510730043671009107920, 3.73890007168283552634154345916, 5.11236438565188037979691475255, 5.86158637769602707959849813028, 7.09615224117569844590414103416, 7.977538146707427006963364723947, 8.969226366586102653189417189460, 9.630680037921911745300815262706, 10.61361711283962193725652179308

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