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Properties

Label 2-735-105.104-c1-0-20
Degree $2$
Conductor $735$
Sign $-0.684 - 0.729i$
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  + 0.644·2-s + (0.536 + 1.64i)3-s − 1.58·4-s + (2.15 + 0.607i)5-s + (0.346 + 1.06i)6-s − 2.31·8-s + (−2.42 + 1.76i)9-s + (1.38 + 0.391i)10-s + 4.05i·11-s + (−0.850 − 2.60i)12-s − 4.21·13-s + (0.155 + 3.86i)15-s + 1.67·16-s − 2.17i·17-s + (−1.56 + 1.14i)18-s + 4.47i·19-s + ⋯
L(s)  = 1  + 0.455·2-s + (0.310 + 0.950i)3-s − 0.792·4-s + (0.962 + 0.271i)5-s + (0.141 + 0.433i)6-s − 0.817·8-s + (−0.807 + 0.589i)9-s + (0.438 + 0.123i)10-s + 1.22i·11-s + (−0.245 − 0.753i)12-s − 1.16·13-s + (0.0401 + 0.999i)15-s + 0.419·16-s − 0.527i·17-s + (−0.368 + 0.268i)18-s + 1.02i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.684 - 0.729i$
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: \((2,\ 735,\ (\ :1/2),\ -0.684 - 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.611110 + 1.41203i\)
\(L(\frac12)\) \(\approx\) \(0.611110 + 1.41203i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.536 - 1.64i)T \)
5 \( 1 + (-2.15 - 0.607i)T \)
7 \( 1 \)
good2 \( 1 - 0.644T + 2T^{2} \)
11 \( 1 - 4.05iT - 11T^{2} \)
13 \( 1 + 4.21T + 13T^{2} \)
17 \( 1 + 2.17iT - 17T^{2} \)
19 \( 1 - 4.47iT - 19T^{2} \)
23 \( 1 + 0.644T + 23T^{2} \)
29 \( 1 - 1.16iT - 29T^{2} \)
31 \( 1 + 0.391iT - 31T^{2} \)
37 \( 1 - 4.26iT - 37T^{2} \)
41 \( 1 - 2.27T + 41T^{2} \)
43 \( 1 - 6.54iT - 43T^{2} \)
47 \( 1 + 7.80iT - 47T^{2} \)
53 \( 1 + 7.20T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 6.99iT - 61T^{2} \)
67 \( 1 - 8.73iT - 67T^{2} \)
71 \( 1 - 8.13iT - 71T^{2} \)
73 \( 1 + 5.23T + 73T^{2} \)
79 \( 1 - 3.75T + 79T^{2} \)
83 \( 1 + 5.27iT - 83T^{2} \)
89 \( 1 - 0.894T + 89T^{2} \)
97 \( 1 - 3.89T + 97T^{2} \)
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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

−10.15901306767498091889301694483, −9.920426354625421492405424013442, −9.318140337610507953553284680677, −8.325184725987798931829836350726, −7.20340311434937575967554733553, −5.93061397524983602099253303915, −5.05714910706477574558299502105, −4.50572371344609838748167785043, −3.29548515126317271709063436939, −2.20195029353467334312812747010, 0.65282474472271208398855588408, 2.29102442576528286214740529021, 3.31667280650340722559005682037, 4.73517375806097446340792741594, 5.67525166537195096890702404523, 6.29372410285359231290882217990, 7.45589285792171261553554106283, 8.528662001022311697497985956953, 9.042322385584872061961864531785, 9.840781572439528174917036957751

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