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Properties

Label 2-735-21.17-c1-0-17
Degree $2$
Conductor $735$
Sign $-0.190 - 0.981i$
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  + (−2.01 + 1.16i)2-s + (1.21 + 1.23i)3-s + (1.71 − 2.97i)4-s + (−0.5 − 0.866i)5-s + (−3.89 − 1.07i)6-s + 3.33i·8-s + (−0.0404 + 2.99i)9-s + (2.01 + 1.16i)10-s + (2.42 + 1.39i)11-s + (5.75 − 1.50i)12-s − 3.20i·13-s + (0.459 − 1.66i)15-s + (−0.459 − 0.795i)16-s + (0.440 − 0.763i)17-s + (−3.41 − 6.10i)18-s + (−1.90 + 1.09i)19-s + ⋯
L(s)  = 1  + (−1.42 + 0.824i)2-s + (0.702 + 0.711i)3-s + (0.858 − 1.48i)4-s + (−0.223 − 0.387i)5-s + (−1.58 − 0.437i)6-s + 1.18i·8-s + (−0.0134 + 0.999i)9-s + (0.638 + 0.368i)10-s + (0.729 + 0.421i)11-s + (1.66 − 0.433i)12-s − 0.888i·13-s + (0.118 − 0.431i)15-s + (−0.114 − 0.198i)16-s + (0.106 − 0.185i)17-s + (−0.804 − 1.43i)18-s + (−0.436 + 0.251i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.190 - 0.981i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.190 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.595329 + 0.721969i\)
\(L(\frac12)\) \(\approx\) \(0.595329 + 0.721969i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.21 - 1.23i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (2.01 - 1.16i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-2.42 - 1.39i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.20iT - 13T^{2} \)
17 \( 1 + (-0.440 + 0.763i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.90 - 1.09i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.53 + 3.77i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.15iT - 29T^{2} \)
31 \( 1 + (-7.62 - 4.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.203 + 0.352i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.55T + 41T^{2} \)
43 \( 1 + 0.118T + 43T^{2} \)
47 \( 1 + (-1.31 - 2.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.46 - 3.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.04 - 3.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.7 - 6.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.802 + 1.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.25iT - 71T^{2} \)
73 \( 1 + (0.192 + 0.110i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.56 - 2.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.666T + 83T^{2} \)
89 \( 1 + (-0.437 - 0.757i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.37iT - 97T^{2} \)
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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

−10.49457450849976913625904308921, −9.349404614013370607764712943858, −8.961492012794243474343661231159, −8.228754882040086332322000918615, −7.46486066144415292241468959700, −6.58884974577771066685320462185, −5.34318047666484641369403329628, −4.30465831815547654574559119322, −2.91256233140021247629891367188, −1.20172590220257949054379174153, 0.873888480151054599464346343226, 2.08587513206889237162603826331, 3.02317775544125398819292326566, 4.11785668307936204752119175204, 6.17967667487642135653647884420, 7.03636297064954881455385766539, 7.82009950691214535720312648664, 8.586811464719764412290528457286, 9.281704528653825219764409836006, 9.860988341767447080150628511655

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