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Properties

Label 2-735-35.12-c1-0-12
Degree $2$
Conductor $735$
Sign $0.901 - 0.431i$
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  + (−2.03 + 0.544i)2-s + (−0.258 + 0.965i)3-s + (2.10 − 1.21i)4-s + (−2.22 + 0.203i)5-s − 2.10i·6-s + (−0.640 + 0.640i)8-s + (−0.866 − 0.499i)9-s + (4.41 − 1.62i)10-s + (1.33 + 2.31i)11-s + (0.629 + 2.34i)12-s + (−1.22 − 1.22i)13-s + (0.379 − 2.20i)15-s + (−1.47 + 2.55i)16-s + (−6.48 − 1.73i)17-s + (2.03 + 0.544i)18-s + (3.00 − 5.21i)19-s + ⋯
L(s)  = 1  + (−1.43 + 0.385i)2-s + (−0.149 + 0.557i)3-s + (1.05 − 0.607i)4-s + (−0.995 + 0.0912i)5-s − 0.859i·6-s + (−0.226 + 0.226i)8-s + (−0.288 − 0.166i)9-s + (1.39 − 0.514i)10-s + (0.402 + 0.697i)11-s + (0.181 + 0.677i)12-s + (−0.340 − 0.340i)13-s + (0.0979 − 0.568i)15-s + (−0.369 + 0.639i)16-s + (−1.57 − 0.421i)17-s + (0.479 + 0.128i)18-s + (0.690 − 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.901 - 0.431i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.901 - 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.441045 + 0.100127i\)
\(L(\frac12)\) \(\approx\) \(0.441045 + 0.100127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (2.22 - 0.203i)T \)
7 \( 1 \)
good2 \( 1 + (2.03 - 0.544i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-1.33 - 2.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.22 + 1.22i)T + 13iT^{2} \)
17 \( 1 + (6.48 + 1.73i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.00 + 5.21i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0643 + 0.239i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.304iT - 29T^{2} \)
31 \( 1 + (-6.28 + 3.62i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.00 + 0.269i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.05iT - 41T^{2} \)
43 \( 1 + (-0.304 + 0.304i)T - 43iT^{2} \)
47 \( 1 + (0.203 + 0.760i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.81 - 1.82i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.99 + 6.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.79 - 2.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.25 - 4.68i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (3.66 - 13.6i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.78 - 5.64i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.88 - 4.88i)T + 83iT^{2} \)
89 \( 1 + (-3.45 + 5.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.84 + 8.84i)T - 97iT^{2} \)
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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

−10.18009306558034833071515079588, −9.489225565352498081268821035023, −8.797155239716087228978163974299, −7.990947004135769817036998826007, −7.11578068586339088080196849664, −6.54862210629349402110068476521, −4.87841894484535602460229666559, −4.13109429087230066603918519738, −2.57446692974971347152763666153, −0.59728747481386606212696251923, 0.798828188844792395196980466776, 2.11503581043494242109323577783, 3.52619719703344479558297180704, 4.80863812387152059780384615355, 6.30294319269541914478475712283, 7.16938834608174997234393518886, 7.964989475777345695373141787389, 8.603949350656762588278531645535, 9.254957694045910056290156225982, 10.42939882616947354843578925233

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