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Properties

Label 2-735-49.36-c1-0-21
Degree $2$
Conductor $735$
Sign $0.999 + 0.0165i$
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  + (−0.105 − 0.460i)2-s + (0.623 + 0.781i)3-s + (1.60 − 0.771i)4-s + (0.623 + 0.781i)5-s + (0.294 − 0.369i)6-s + (−2.25 − 1.39i)7-s + (−1.11 − 1.39i)8-s + (−0.222 + 0.974i)9-s + (0.294 − 0.369i)10-s + (1.04 + 4.56i)11-s + (1.60 + 0.771i)12-s + (0.142 + 0.623i)13-s + (−0.403 + 1.18i)14-s + (−0.222 + 0.974i)15-s + (1.69 − 2.12i)16-s + (2.90 + 1.39i)17-s + ⋯
L(s)  = 1  + (−0.0742 − 0.325i)2-s + (0.359 + 0.451i)3-s + (0.800 − 0.385i)4-s + (0.278 + 0.349i)5-s + (0.120 − 0.150i)6-s + (−0.850 − 0.525i)7-s + (−0.393 − 0.492i)8-s + (−0.0741 + 0.324i)9-s + (0.0930 − 0.116i)10-s + (0.313 + 1.37i)11-s + (0.462 + 0.222i)12-s + (0.0394 + 0.172i)13-s + (−0.107 + 0.315i)14-s + (−0.0574 + 0.251i)15-s + (0.422 − 0.530i)16-s + (0.704 + 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.999 + 0.0165i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (526, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.999 + 0.0165i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03432 - 0.0168692i\)
\(L(\frac12)\) \(\approx\) \(2.03432 - 0.0168692i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.623 - 0.781i)T \)
5 \( 1 + (-0.623 - 0.781i)T \)
7 \( 1 + (2.25 + 1.39i)T \)
good2 \( 1 + (0.105 + 0.460i)T + (-1.80 + 0.867i)T^{2} \)
11 \( 1 + (-1.04 - 4.56i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.142 - 0.623i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-2.90 - 1.39i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 - 8.07T + 19T^{2} \)
23 \( 1 + (-7.66 + 3.69i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (-0.216 - 0.104i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 3.02T + 31T^{2} \)
37 \( 1 + (7.07 + 3.40i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-3.23 - 4.05i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-0.0362 + 0.0454i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (0.410 + 1.80i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-1.50 + 0.726i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-1.86 + 2.33i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (8.53 + 4.10i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + 15.6T + 67T^{2} \)
71 \( 1 + (10.7 - 5.17i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-2.95 + 12.9i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + (3.07 - 13.4i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (0.0120 - 0.0527i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−10.27912766131637605465676692091, −9.703042718266920338237986438940, −9.114954939997587591343610968107, −7.37141803570647728265220295477, −7.11806300450954308909082098797, −6.02196776499603777513865638469, −4.90905872601791633134579371782, −3.54404529236496904385407762750, −2.80340614861400570806186910927, −1.42480787025848217506093687587, 1.26676568925972573799009921016, 3.04071083099275277938334365111, 3.24861409540024308061056999059, 5.46709809940542005442166158074, 5.91828518921764037126155783369, 7.04013028852719495342181667291, 7.61500775815346368318068187644, 8.796952972634559540663982951381, 9.148097324942187794529253721707, 10.33063418397588565269610397713

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