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Properties

Label 2-735-15.2-c1-0-49
Degree $2$
Conductor $735$
Sign $0.105 + 0.994i$
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  + (−0.347 + 0.347i)2-s + (−1.72 + 0.176i)3-s + 1.75i·4-s + (1.16 − 1.90i)5-s + (0.536 − 0.659i)6-s + (−1.30 − 1.30i)8-s + (2.93 − 0.607i)9-s + (0.256 + 1.06i)10-s − 2.67i·11-s + (−0.310 − 3.03i)12-s + (−2.14 + 2.14i)13-s + (−1.67 + 3.49i)15-s − 2.61·16-s + (−3.26 + 3.26i)17-s + (−0.808 + 1.23i)18-s − 5.24i·19-s + ⋯
L(s)  = 1  + (−0.245 + 0.245i)2-s + (−0.994 + 0.101i)3-s + 0.879i·4-s + (0.522 − 0.852i)5-s + (0.219 − 0.269i)6-s + (−0.461 − 0.461i)8-s + (0.979 − 0.202i)9-s + (0.0810 + 0.337i)10-s − 0.805i·11-s + (−0.0895 − 0.874i)12-s + (−0.596 + 0.596i)13-s + (−0.432 + 0.901i)15-s − 0.653·16-s + (−0.792 + 0.792i)17-s + (−0.190 + 0.290i)18-s − 1.20i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.105 + 0.994i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.105 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.418123 - 0.375978i\)
\(L(\frac12)\) \(\approx\) \(0.418123 - 0.375978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.176i)T \)
5 \( 1 + (-1.16 + 1.90i)T \)
7 \( 1 \)
good2 \( 1 + (0.347 - 0.347i)T - 2iT^{2} \)
11 \( 1 + 2.67iT - 11T^{2} \)
13 \( 1 + (2.14 - 2.14i)T - 13iT^{2} \)
17 \( 1 + (3.26 - 3.26i)T - 17iT^{2} \)
19 \( 1 + 5.24iT - 19T^{2} \)
23 \( 1 + (2.54 + 2.54i)T + 23iT^{2} \)
29 \( 1 + 2.86T + 29T^{2} \)
31 \( 1 - 5.28T + 31T^{2} \)
37 \( 1 + (2.14 + 2.14i)T + 37iT^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
43 \( 1 + (-0.759 + 0.759i)T - 43iT^{2} \)
47 \( 1 + (-7.66 + 7.66i)T - 47iT^{2} \)
53 \( 1 + (-4.43 - 4.43i)T + 53iT^{2} \)
59 \( 1 + 0.159T + 59T^{2} \)
61 \( 1 + 4.72T + 61T^{2} \)
67 \( 1 + (5.41 + 5.41i)T + 67iT^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 + (4.16 - 4.16i)T - 73iT^{2} \)
79 \( 1 - 3.89iT - 79T^{2} \)
83 \( 1 + (4.03 + 4.03i)T + 83iT^{2} \)
89 \( 1 + 3.95T + 89T^{2} \)
97 \( 1 + (-1.86 - 1.86i)T + 97iT^{2} \)
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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

−10.20087207495104577186903033842, −9.046135825944509473101942387477, −8.735904035293409747992592775220, −7.48174661923765811040069089291, −6.61998733790214230865772076943, −5.81899316647893168515505035788, −4.71919139998742854208424726093, −3.95590669530464450471968270962, −2.21949528371121479272939433974, −0.35396190936899233767671025655, 1.50014452966456527595009991582, 2.61837418376311915715882008858, 4.41737730825946140220799060337, 5.40225076432054495049759634146, 6.10290285509235309654908724841, 6.90092492891543027642708027439, 7.76566247695728355457637419582, 9.367711650047738443455885997907, 10.03707088130604415547284860559, 10.36741832501820239200080044495

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