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Properties

Label 2-735-7.2-c1-0-25
Degree $2$
Conductor $735$
Sign $-0.386 - 0.922i$
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  + (1.36 − 2.36i)2-s + (−0.5 − 0.866i)3-s + (−2.73 − 4.73i)4-s + (0.5 − 0.866i)5-s − 2.73·6-s − 9.46·8-s + (−0.499 + 0.866i)9-s + (−1.36 − 2.36i)10-s + (−0.366 − 0.633i)11-s + (−2.73 + 4.73i)12-s − 2.26·13-s − 0.999·15-s + (−7.46 + 12.9i)16-s + (1.63 + 2.83i)17-s + (1.36 + 2.36i)18-s + (2.23 − 3.86i)19-s + ⋯
L(s)  = 1  + (0.965 − 1.67i)2-s + (−0.288 − 0.499i)3-s + (−1.36 − 2.36i)4-s + (0.223 − 0.387i)5-s − 1.11·6-s − 3.34·8-s + (−0.166 + 0.288i)9-s + (−0.431 − 0.748i)10-s + (−0.110 − 0.191i)11-s + (−0.788 + 1.36i)12-s − 0.629·13-s − 0.258·15-s + (−1.86 + 3.23i)16-s + (0.396 + 0.686i)17-s + (0.321 + 0.557i)18-s + (0.512 − 0.886i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.936654 + 1.40811i\)
\(L(\frac12)\) \(\approx\) \(0.936654 + 1.40811i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-1.36 + 2.36i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (0.366 + 0.633i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.26T + 13T^{2} \)
17 \( 1 + (-1.63 - 2.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.23 + 3.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.36 + 4.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.19T + 29T^{2} \)
31 \( 1 + (0.232 + 0.401i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.59 + 2.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.732T + 41T^{2} \)
43 \( 1 - 3.19T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.19 + 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.0980 + 0.169i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.33 - 12.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.19T + 71T^{2} \)
73 \( 1 + (-6.33 - 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.69 + 6.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 + (-7.56 + 13.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−10.03655412739056655578071287813, −9.359280590491839749417667862258, −8.354210578383913073998132592962, −6.87725441532551793715286833340, −5.72623689552205580185826978586, −5.11113846722147518002436860972, −4.13272529985834546322829830060, −2.92339452240120269159374784021, −1.94367666901155453145104409604, −0.66268880097758589951050033581, 2.97678257465994527447304237680, 3.91827784839257335120135753697, 5.00015481438260493566659400947, 5.56115187960379510939829901720, 6.45120218555757878650516081746, 7.38660134583979854023875701144, 7.904312830946939254882381723741, 9.217740949723812143987083592816, 9.742406812910658278806732703547, 11.10925669291739991119649671083

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