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Properties

Label 2-435-145.102-c1-0-11
Degree $2$
Conductor $435$
Sign $-0.676 + 0.736i$
Analytic cond. $3.47349$
Root an. cond. $1.86373$
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  + (−2.08 − 1.66i)2-s + (0.222 − 0.974i)3-s + (1.14 + 5.00i)4-s + (−0.847 − 2.06i)5-s + (−2.08 + 1.66i)6-s + (2.44 − 1.53i)7-s + (3.63 − 7.54i)8-s + (−0.900 − 0.433i)9-s + (−1.67 + 5.73i)10-s + (5.75 + 2.01i)11-s + 5.13·12-s + (5.07 + 1.77i)13-s + (−7.67 − 0.865i)14-s + (−2.20 + 0.366i)15-s + (−10.9 + 5.25i)16-s − 1.01i·17-s + ⋯
L(s)  = 1  + (−1.47 − 1.17i)2-s + (0.128 − 0.562i)3-s + (0.571 + 2.50i)4-s + (−0.379 − 0.925i)5-s + (−0.852 + 0.679i)6-s + (0.925 − 0.581i)7-s + (1.28 − 2.66i)8-s + (−0.300 − 0.144i)9-s + (−0.529 + 1.81i)10-s + (1.73 + 0.607i)11-s + 1.48·12-s + (1.40 + 0.492i)13-s + (−2.05 − 0.231i)14-s + (−0.569 + 0.0945i)15-s + (−2.72 + 1.31i)16-s − 0.246i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-0.676 + 0.736i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ -0.676 + 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337799 - 0.769660i\)
\(L(\frac12)\) \(\approx\) \(0.337799 - 0.769660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.847 + 2.06i)T \)
29 \( 1 + (3.41 + 4.16i)T \)
good2 \( 1 + (2.08 + 1.66i)T + (0.445 + 1.94i)T^{2} \)
7 \( 1 + (-2.44 + 1.53i)T + (3.03 - 6.30i)T^{2} \)
11 \( 1 + (-5.75 - 2.01i)T + (8.60 + 6.85i)T^{2} \)
13 \( 1 + (-5.07 - 1.77i)T + (10.1 + 8.10i)T^{2} \)
17 \( 1 + 1.01iT - 17T^{2} \)
19 \( 1 + (-0.508 - 0.319i)T + (8.24 + 17.1i)T^{2} \)
23 \( 1 + (-5.33 - 0.600i)T + (22.4 + 5.11i)T^{2} \)
31 \( 1 + (-6.69 + 0.753i)T + (30.2 - 6.89i)T^{2} \)
37 \( 1 + (2.41 + 1.16i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (6.09 - 6.09i)T - 41iT^{2} \)
43 \( 1 + (6.57 + 8.24i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (6.94 - 3.34i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.344 - 3.05i)T + (-51.6 + 11.7i)T^{2} \)
59 \( 1 - 7.34iT - 59T^{2} \)
61 \( 1 + (0.588 - 0.369i)T + (26.4 - 54.9i)T^{2} \)
67 \( 1 + (2.74 - 0.959i)T + (52.3 - 41.7i)T^{2} \)
71 \( 1 + (0.234 + 0.486i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-4.42 + 3.52i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (1.59 - 0.558i)T + (61.7 - 49.2i)T^{2} \)
83 \( 1 + (-2.95 - 1.85i)T + (36.0 + 74.7i)T^{2} \)
89 \( 1 + (1.32 + 11.7i)T + (-86.7 + 19.8i)T^{2} \)
97 \( 1 + (0.130 + 0.569i)T + (-87.3 + 42.0i)T^{2} \)
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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

−10.99877065323691766202805253149, −9.738668743525330571805875736755, −8.934314199299727533391801202851, −8.431707911991903820988028785361, −7.55130329006897372706751296496, −6.62955866136781156962087999665, −4.45828241763287052153386121838, −3.57012302284836541778241711647, −1.62102420853638248537544124704, −1.14005598311501528702387411409, 1.42941678427147307244596696210, 3.48800151802092460477035845316, 5.14746502317943757710848452808, 6.26986593718361710671783370246, 6.82447551355085537637320343260, 8.204846411898008576923084276594, 8.530574173602201844300389718136, 9.368828276884017160802107608821, 10.43442161789557722117965322691, 11.17870537299501751259765331099

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