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Properties

Label 2-735-5.4-c1-0-0
Degree $2$
Conductor $735$
Sign $-0.774 + 0.632i$
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  + 1.93i·2-s − i·3-s − 1.73·4-s + (−1.41 − 1.73i)5-s + 1.93·6-s + 0.517i·8-s − 9-s + (3.34 − 2.73i)10-s − 3.46·11-s + 1.73i·12-s + 4i·13-s + (−1.73 + 1.41i)15-s − 4.46·16-s − 4i·17-s − 1.93i·18-s + 0.378·19-s + ⋯
L(s)  = 1  + 1.36i·2-s − 0.577i·3-s − 0.866·4-s + (−0.632 − 0.774i)5-s + 0.788·6-s + 0.183i·8-s − 0.333·9-s + (1.05 − 0.863i)10-s − 1.04·11-s + 0.500i·12-s + 1.10i·13-s + (−0.447 + 0.365i)15-s − 1.11·16-s − 0.970i·17-s − 0.455i·18-s + 0.0869·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.774 + 0.632i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0790825 - 0.221896i\)
\(L(\frac12)\) \(\approx\) \(0.0790825 - 0.221896i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (1.41 + 1.73i)T \)
7 \( 1 \)
good2 \( 1 - 1.93iT - 2T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 0.378T + 19T^{2} \)
23 \( 1 - 6.31iT - 23T^{2} \)
29 \( 1 + 8.92T + 29T^{2} \)
31 \( 1 + 7.34T + 31T^{2} \)
37 \( 1 - 0.757iT - 37T^{2} \)
41 \( 1 + 8.48T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 7.34iT - 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 9.14T + 61T^{2} \)
67 \( 1 - 6.96iT - 67T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 - 10.9iT - 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 4.14T + 89T^{2} \)
97 \( 1 + 5.07iT - 97T^{2} \)
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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

−11.27151899836437135539584682996, −9.582095219455740719251126434529, −8.892245817847791756784018369502, −8.018088396196062562609608116977, −7.39537109112062604013721237980, −6.82625601484252429815580011616, −5.44426418218221373832655456073, −5.13201808618868524159134312199, −3.72619546432153492796452819380, −1.97164062563297937715560680232, 0.11230493458551007059891786332, 2.19702728724732113146310603719, 3.19872095737642322458585764762, 3.82613793257231242597309091962, 4.99953448077176096315822396096, 6.19349935683189189925694159750, 7.46261548119238737127561400691, 8.257535580421499432557463101708, 9.367316094136617471271826948978, 10.33590345762475132725526069594

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