Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $-0.949 + 0.314i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.544 − 2.03i)2-s + (0.965 − 0.258i)3-s + (−2.10 − 1.21i)4-s + (0.936 − 2.03i)5-s − 2.10i·6-s + (−0.640 + 0.640i)8-s + (0.866 − 0.499i)9-s + (−3.61 − 3.01i)10-s + (1.33 − 2.31i)11-s + (−2.34 − 0.629i)12-s + (−1.22 − 1.22i)13-s + (0.379 − 2.20i)15-s + (−1.47 − 2.55i)16-s + (1.73 + 6.48i)17-s + (−0.544 − 2.03i)18-s + (3.00 + 5.21i)19-s + ⋯
L(s)  = 1  + (0.385 − 1.43i)2-s + (0.557 − 0.149i)3-s + (−1.05 − 0.607i)4-s + (0.418 − 0.908i)5-s − 0.859i·6-s + (−0.226 + 0.226i)8-s + (0.288 − 0.166i)9-s + (−1.14 − 0.952i)10-s + (0.402 − 0.697i)11-s + (−0.677 − 0.181i)12-s + (−0.340 − 0.340i)13-s + (0.0979 − 0.568i)15-s + (−0.369 − 0.639i)16-s + (0.421 + 1.57i)17-s + (−0.128 − 0.479i)18-s + (0.690 + 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.949 + 0.314i$
motivic weight  =  \(1\)
character  :  $\chi_{735} (472, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ -0.949 + 0.314i)\)
\(L(1)\)  \(\approx\)  \(0.381954 - 2.36885i\)
\(L(\frac12)\)  \(\approx\)  \(0.381954 - 2.36885i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-0.936 + 2.03i)T \)
7 \( 1 \)
good2 \( 1 + (-0.544 + 2.03i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-1.33 + 2.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.22 + 1.22i)T + 13iT^{2} \)
17 \( 1 + (-1.73 - 6.48i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.00 - 5.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.239 - 0.0643i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.304iT - 29T^{2} \)
31 \( 1 + (6.28 + 3.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.269 - 1.00i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.05iT - 41T^{2} \)
43 \( 1 + (-0.304 + 0.304i)T - 43iT^{2} \)
47 \( 1 + (-0.760 - 0.203i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.82 + 6.81i)T + (-45.8 + 26.5i)T^{2} \)
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 \( 1 + (-4.68 + 1.25i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (-13.6 + 3.66i)T + (63.2 - 36.5i)T^{2} \)
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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.985196369394118617109506910348, −9.469186079389810412324661316613, −8.472249424450338387788470641596, −7.73643854345873753332755900626, −6.16636901455866469657226411785, −5.24688649318170152265290802932, −4.02587535509857664892743449769, −3.37860210556023686577823970220, −2.02174622389820115689942521035, −1.14626395586208455604178708357, 2.21400179551503580139420895483, 3.42779511471687460543789408952, 4.72790978481701945824404974070, 5.42349367475648868622571090886, 6.69869861349573979000372724252, 7.14164067272836079023042100200, 7.73014726486895890852501732787, 9.156773420988536925530123125018, 9.451650273206634122316294873097, 10.65945512395279850508643846126

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