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Properties

Label 2-735-105.2-c1-0-56
Degree $2$
Conductor $735$
Sign $-0.723 + 0.689i$
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  + (−0.455 − 1.69i)2-s + (1.24 + 1.20i)3-s + (−0.948 + 0.547i)4-s + (−0.445 − 2.19i)5-s + (1.48 − 2.66i)6-s + (−1.12 − 1.12i)8-s + (0.0946 + 2.99i)9-s + (−3.52 + 1.75i)10-s + (1.34 − 0.776i)11-s + (−1.84 − 0.462i)12-s + (4.50 − 4.50i)13-s + (2.08 − 3.26i)15-s + (−2.49 + 4.32i)16-s + (−2.91 − 0.780i)17-s + (5.05 − 1.52i)18-s + (−3.64 − 2.10i)19-s + ⋯
L(s)  = 1  + (−0.322 − 1.20i)2-s + (0.718 + 0.695i)3-s + (−0.474 + 0.273i)4-s + (−0.199 − 0.979i)5-s + (0.604 − 1.08i)6-s + (−0.397 − 0.397i)8-s + (0.0315 + 0.999i)9-s + (−1.11 + 0.554i)10-s + (0.405 − 0.234i)11-s + (−0.531 − 0.133i)12-s + (1.25 − 1.25i)13-s + (0.539 − 0.842i)15-s + (−0.623 + 1.08i)16-s + (−0.706 − 0.189i)17-s + (1.19 − 0.359i)18-s + (−0.836 − 0.482i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.723 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.723 + 0.689i$
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: \((2,\ 735,\ (\ :1/2),\ -0.723 + 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.568038 - 1.41925i\)
\(L(\frac12)\) \(\approx\) \(0.568038 - 1.41925i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.24 - 1.20i)T \)
5 \( 1 + (0.445 + 2.19i)T \)
7 \( 1 \)
good2 \( 1 + (0.455 + 1.69i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-1.34 + 0.776i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.50 + 4.50i)T - 13iT^{2} \)
17 \( 1 + (2.91 + 0.780i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.64 + 2.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.13 + 1.37i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.97T + 29T^{2} \)
31 \( 1 + (2.89 + 5.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.68 + 0.450i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.68iT - 41T^{2} \)
43 \( 1 + (2.09 - 2.09i)T - 43iT^{2} \)
47 \( 1 + (-0.0131 - 0.0489i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.57 + 5.88i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.46 + 4.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.65 + 2.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.625 + 2.33i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.73iT - 71T^{2} \)
73 \( 1 + (-9.92 - 2.66i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.11 - 1.79i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.2 - 12.2i)T + 83iT^{2} \)
89 \( 1 + (0.678 - 1.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.9 + 10.9i)T + 97iT^{2} \)
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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

−10.08106638443842915819698040241, −9.194198657406168236153627097816, −8.718245038527612841869577189101, −8.033507976652940361147280414088, −6.47908246607389103790129009648, −5.22035513417186068578087285254, −4.13275931098420636454261124315, −3.38152010155454314697361680947, −2.23870018332078664321764937985, −0.824278620249291190969892014237, 1.84657442676881811768379618481, 3.12714041690235635924870402293, 4.23636211177430010894559447757, 5.99637266535984101328839403729, 6.69513025957132753233882141886, 6.99346791394900581268928614689, 8.036001032762418951717923548559, 8.758290128359780458738112612247, 9.328633231039014950919849930129, 10.72930846393460033962900313315

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