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Properties

Label 2-735-21.20-c1-0-14
Degree $2$
Conductor $735$
Sign $-0.932 - 0.360i$
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  + 1.07i·2-s + (0.812 + 1.52i)3-s + 0.837·4-s − 5-s + (−1.64 + 0.876i)6-s + 3.05i·8-s + (−1.68 + 2.48i)9-s − 1.07i·10-s + 4.43i·11-s + (0.680 + 1.28i)12-s − 0.955i·13-s + (−0.812 − 1.52i)15-s − 1.62·16-s + 0.507·17-s + (−2.68 − 1.81i)18-s − 5.09i·19-s + ⋯
L(s)  = 1  + 0.762i·2-s + (0.469 + 0.883i)3-s + 0.418·4-s − 0.447·5-s + (−0.673 + 0.357i)6-s + 1.08i·8-s + (−0.560 + 0.828i)9-s − 0.341i·10-s + 1.33i·11-s + (0.196 + 0.369i)12-s − 0.265i·13-s + (−0.209 − 0.394i)15-s − 0.406·16-s + 0.123·17-s + (−0.631 − 0.427i)18-s − 1.16i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.932 - 0.360i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.932 - 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324137 + 1.73740i\)
\(L(\frac12)\) \(\approx\) \(0.324137 + 1.73740i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.812 - 1.52i)T \)
5 \( 1 + T \)
7 \( 1 \)
good2 \( 1 - 1.07iT - 2T^{2} \)
11 \( 1 - 4.43iT - 11T^{2} \)
13 \( 1 + 0.955iT - 13T^{2} \)
17 \( 1 - 0.507T + 17T^{2} \)
19 \( 1 + 5.09iT - 19T^{2} \)
23 \( 1 + 4.29iT - 23T^{2} \)
29 \( 1 - 6.89iT - 29T^{2} \)
31 \( 1 - 5.89iT - 31T^{2} \)
37 \( 1 - 7.52T + 37T^{2} \)
41 \( 1 + 4.65T + 41T^{2} \)
43 \( 1 + 0.492T + 43T^{2} \)
47 \( 1 + 6.65T + 47T^{2} \)
53 \( 1 + 9.13iT - 53T^{2} \)
59 \( 1 - 11.6T + 59T^{2} \)
61 \( 1 + 0.461iT - 61T^{2} \)
67 \( 1 + 3.70T + 67T^{2} \)
71 \( 1 - 7.90iT - 71T^{2} \)
73 \( 1 + 6.31iT - 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 7.15T + 89T^{2} \)
97 \( 1 + 6.91iT - 97T^{2} \)
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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

−10.67728195651029396452134802296, −9.882449586754590525310944119134, −8.864923979962289990983405918484, −8.171281197539830277453732546020, −7.26421900142334005870625479846, −6.59258106214249481935429795006, −5.15121945532637977275419804153, −4.67313259121661904120417563545, −3.28504004720751789896922031659, −2.21032009329386960814702560360, 0.854489920890950475411380147822, 2.11216213110222057231871416283, 3.22073019349252472914499482913, 3.91750363899646854709269692980, 5.83991793735727122444079261047, 6.44606841762200353802051412069, 7.63907784657507275442711533548, 8.081102945982974401497151905321, 9.210175919103517913825493166837, 10.06052122141566721410466295421

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