Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.812 + 0.583i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.540 − 0.540i)2-s + (0.707 − 0.707i)3-s + 1.41i·4-s + (−1.03 − 1.98i)5-s − 0.763i·6-s + (2.57 − 0.614i)7-s + (1.84 + 1.84i)8-s − 1.00i·9-s + (−1.63 − 0.510i)10-s − 3.85·11-s + (1.00 + 1.00i)12-s + (−3.66 + 3.66i)13-s + (1.05 − 1.72i)14-s + (−2.13 − 0.668i)15-s − 0.839·16-s + (1.49 + 1.49i)17-s + ⋯
L(s)  = 1  + (0.381 − 0.381i)2-s + (0.408 − 0.408i)3-s + 0.708i·4-s + (−0.463 − 0.886i)5-s − 0.311i·6-s + (0.972 − 0.232i)7-s + (0.652 + 0.652i)8-s − 0.333i·9-s + (−0.515 − 0.161i)10-s − 1.16·11-s + (0.289 + 0.289i)12-s + (−1.01 + 1.01i)13-s + (0.282 − 0.460i)14-s + (−0.550 − 0.172i)15-s − 0.209·16-s + (0.361 + 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.812 + 0.583i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1/2),\ 0.812 + 0.583i)\)
\(L(1)\)  \(\approx\)  \(1.24914 - 0.402038i\)
\(L(\frac12)\)  \(\approx\)  \(1.24914 - 0.402038i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (1.03 + 1.98i)T \)
7 \( 1 + (-2.57 + 0.614i)T \)
good2 \( 1 + (-0.540 + 0.540i)T - 2iT^{2} \)
11 \( 1 + 3.85T + 11T^{2} \)
13 \( 1 + (3.66 - 3.66i)T - 13iT^{2} \)
17 \( 1 + (-1.49 - 1.49i)T + 17iT^{2} \)
19 \( 1 - 0.0697T + 19T^{2} \)
23 \( 1 + (0.534 + 0.534i)T + 23iT^{2} \)
29 \( 1 + 2.77iT - 29T^{2} \)
31 \( 1 - 2.39iT - 31T^{2} \)
37 \( 1 + (-6.18 + 6.18i)T - 37iT^{2} \)
41 \( 1 + 8.68iT - 41T^{2} \)
43 \( 1 + (2.77 + 2.77i)T + 43iT^{2} \)
47 \( 1 + (-5.49 - 5.49i)T + 47iT^{2} \)
53 \( 1 + (-6.13 - 6.13i)T + 53iT^{2} \)
59 \( 1 + 6.97T + 59T^{2} \)
61 \( 1 + 14.3iT - 61T^{2} \)
67 \( 1 + (-0.416 + 0.416i)T - 67iT^{2} \)
71 \( 1 + 8.12T + 71T^{2} \)
73 \( 1 + (-9.55 + 9.55i)T - 73iT^{2} \)
79 \( 1 - 9.86iT - 79T^{2} \)
83 \( 1 + (-1.63 + 1.63i)T - 83iT^{2} \)
89 \( 1 + 5.05T + 89T^{2} \)
97 \( 1 + (-6.85 - 6.85i)T + 97iT^{2} \)
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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

−13.53049574492770267778626534289, −12.50069019940658865073547236293, −11.95759074583157083079980706283, −10.80646031671678837149610147589, −9.091442861830816726274415863150, −7.979236857424439912323039164949, −7.46834052262121055454367194578, −5.10010336689761013924167855500, −4.07991046757765507334050106279, −2.21015549115096211005793786033, 2.72972439489513917433532881784, 4.63607480728267832125508198744, 5.60915817274846343982356237311, 7.31319864194481956553868314226, 8.097694927807867396376461256378, 9.909525585556248172347520835524, 10.54356611686494288994117337407, 11.61417334397600777814516906832, 13.15614665906510407811310462564, 14.22550512591727065934597684106

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