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Properties

Label 2-735-35.9-c1-0-3
Degree $2$
Conductor $735$
Sign $-0.0530 - 0.998i$
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  + (−0.167 − 0.0969i)2-s + (−0.866 + 0.5i)3-s + (−0.981 − 1.69i)4-s + (0.710 + 2.12i)5-s + 0.193·6-s + 0.768i·8-s + (0.499 − 0.866i)9-s + (0.0863 − 0.424i)10-s + (−1 − 1.73i)11-s + (1.69 + 0.981i)12-s + 1.35i·13-s + (−1.67 − 1.48i)15-s + (−1.88 + 3.26i)16-s + (2.90 − 1.67i)17-s + (−0.167 + 0.0969i)18-s + (−2.67 + 4.63i)19-s + ⋯
L(s)  = 1  + (−0.118 − 0.0685i)2-s + (−0.499 + 0.288i)3-s + (−0.490 − 0.849i)4-s + (0.317 + 0.948i)5-s + 0.0791·6-s + 0.271i·8-s + (0.166 − 0.288i)9-s + (0.0273 − 0.134i)10-s + (−0.301 − 0.522i)11-s + (0.490 + 0.283i)12-s + 0.374i·13-s + (−0.432 − 0.382i)15-s + (−0.471 + 0.817i)16-s + (0.703 − 0.406i)17-s + (−0.0395 + 0.0228i)18-s + (−0.613 + 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0530 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0530 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.0530 - 0.998i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.0530 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.545263 + 0.574995i\)
\(L(\frac12)\) \(\approx\) \(0.545263 + 0.574995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.710 - 2.12i)T \)
7 \( 1 \)
good2 \( 1 + (0.167 + 0.0969i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.35iT - 13T^{2} \)
17 \( 1 + (-2.90 + 1.67i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.67 - 4.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.29 - 2.48i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 + (-2.28 - 3.96i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.671 - 0.387i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.73T + 41T^{2} \)
43 \( 1 - 12.6iT - 43T^{2} \)
47 \( 1 + (-8.59 - 4.96i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.42 - 4.28i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.31 - 7.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.35 - 7.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.59 + 4.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (8.09 - 4.67i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.35 + 9.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.22iT - 83T^{2} \)
89 \( 1 + (0.518 - 0.898i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 18.4iT - 97T^{2} \)
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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

−10.72757136767792082736834366964, −9.777610283381917561749957440362, −9.275012807624256114959890697499, −8.037444072721389945130199026908, −6.93966222862326170665630596347, −5.96420102539826602830682210456, −5.46912195428793685578169341062, −4.26117406494224185433836753535, −3.04264921476603153708907347099, −1.46216831718577683317953241346, 0.47854065465184582465906964037, 2.22375178452815993139127225479, 3.77304459210348617358005750614, 4.81768646876431744169543235644, 5.48782692151772732243347018062, 6.77148119308345587196029098939, 7.64697225260456357133752016734, 8.451423268630439848654224105951, 9.189039776413257005474550751443, 10.03366081013633509325594925875

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