Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $-0.204 + 0.978i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.228 − 0.0611i)2-s + (0.258 − 0.965i)3-s + (−1.68 + 0.972i)4-s + (−1.18 + 1.89i)5-s − 0.236i·6-s + (−0.658 + 0.658i)8-s + (−0.866 − 0.499i)9-s + (−0.155 + 0.504i)10-s + (−1.99 − 3.45i)11-s + (0.503 + 1.87i)12-s + (0.500 + 0.500i)13-s + (1.52 + 1.63i)15-s + (1.83 − 3.17i)16-s + (2.29 + 0.614i)17-s + (−0.228 − 0.0611i)18-s + (3.60 − 6.25i)19-s + ⋯
L(s)  = 1  + (0.161 − 0.0432i)2-s + (0.149 − 0.557i)3-s + (−0.841 + 0.486i)4-s + (−0.532 + 0.846i)5-s − 0.0964i·6-s + (−0.232 + 0.232i)8-s + (−0.288 − 0.166i)9-s + (−0.0492 + 0.159i)10-s + (−0.600 − 1.04i)11-s + (0.145 + 0.542i)12-s + (0.138 + 0.138i)13-s + (0.392 + 0.423i)15-s + (0.458 − 0.794i)16-s + (0.556 + 0.148i)17-s + (−0.0537 − 0.0144i)18-s + (0.828 − 1.43i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.204 + 0.978i$
motivic weight  =  \(1\)
character  :  $\chi_{735} (607, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ -0.204 + 0.978i)\)
\(L(1)\)  \(\approx\)  \(0.495647 - 0.609828i\)
\(L(\frac12)\)  \(\approx\)  \(0.495647 - 0.609828i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (1.18 - 1.89i)T \)
7 \( 1 \)
good2 \( 1 + (-0.228 + 0.0611i)T + (1.73 - i)T^{2} \)
11 \( 1 + (1.99 + 3.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.500 - 0.500i)T + 13iT^{2} \)
17 \( 1 + (-2.29 - 0.614i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.60 + 6.25i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.88 + 7.04i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.65iT - 29T^{2} \)
31 \( 1 + (4.27 - 2.46i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.399 - 0.106i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.63iT - 41T^{2} \)
43 \( 1 + (-3.65 + 3.65i)T - 43iT^{2} \)
47 \( 1 + (0.111 + 0.417i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (7.37 + 1.97i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.05 + 5.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.15 - 3.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.345 + 1.28i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.19T + 71T^{2} \)
73 \( 1 + (-0.506 + 1.88i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.48 + 4.32i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.9 + 11.9i)T + 83iT^{2} \)
89 \( 1 + (3.91 - 6.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.43 + 7.43i)T - 97iT^{2} \)
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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

−10.19253437319323660867152853525, −9.090269439902814751654143208249, −8.277020745795112316988322472529, −7.68791712458154570747905779740, −6.73405326017091432933201142154, −5.69460710158406710191845600255, −4.54089081552070794903572763809, −3.37129114252153732870418137742, −2.71124921939476325410154618147, −0.40744450130136418879187670404, 1.47216825342047502314541192110, 3.50155303364833281014642848620, 4.22432721268435453249544423663, 5.28408851586970157174558706221, 5.62982250604808291166016950467, 7.46485057794216589287227867702, 8.061227166882382297809540544633, 9.120751395798774100076267554038, 9.696666406841608279143747558950, 10.32693065157909823433561443302

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