Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.993 - 0.111i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.67 − 1.67i)2-s + (2.55 + 1.56i)3-s + 1.58i·4-s + (−0.529 + 4.97i)5-s + (−1.66 − 6.89i)6-s + (6.78 + 1.73i)7-s + (−4.03 + 4.03i)8-s + (4.10 + 8.01i)9-s + (9.19 − 7.42i)10-s + 4.41i·11-s + (−2.48 + 4.06i)12-s + (1.62 − 1.62i)13-s + (−8.43 − 14.2i)14-s + (−9.13 + 11.8i)15-s + 19.8·16-s + (13.9 − 13.9i)17-s + ⋯
L(s)  = 1  + (−0.835 − 0.835i)2-s + (0.853 + 0.521i)3-s + 0.397i·4-s + (−0.105 + 0.994i)5-s + (−0.277 − 1.14i)6-s + (0.968 + 0.247i)7-s + (−0.503 + 0.503i)8-s + (0.455 + 0.890i)9-s + (0.919 − 0.742i)10-s + 0.401i·11-s + (−0.207 + 0.338i)12-s + (0.124 − 0.124i)13-s + (−0.602 − 1.01i)14-s + (−0.609 + 0.793i)15-s + 1.23·16-s + (0.819 − 0.819i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.993 - 0.111i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (83, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.993 - 0.111i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.19838 + 0.0670374i\)
\(L(\frac12)\)  \(\approx\)  \(1.19838 + 0.0670374i\)
\(L(2)\)   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 \( 1 + (-2.55 - 1.56i)T \)
5 \( 1 + (0.529 - 4.97i)T \)
7 \( 1 + (-6.78 - 1.73i)T \)
good2 \( 1 + (1.67 + 1.67i)T + 4iT^{2} \)
11 \( 1 - 4.41iT - 121T^{2} \)
13 \( 1 + (-1.62 + 1.62i)T - 169iT^{2} \)
17 \( 1 + (-13.9 + 13.9i)T - 289iT^{2} \)
19 \( 1 + 0.694T + 361T^{2} \)
23 \( 1 + (-23.1 + 23.1i)T - 529iT^{2} \)
29 \( 1 + 49.1T + 841T^{2} \)
31 \( 1 - 33.8iT - 961T^{2} \)
37 \( 1 + (-2.02 + 2.02i)T - 1.36e3iT^{2} \)
41 \( 1 + 32.5T + 1.68e3T^{2} \)
43 \( 1 + (30.4 + 30.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-18.7 + 18.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-33.8 + 33.8i)T - 2.80e3iT^{2} \)
59 \( 1 + 23.1iT - 3.48e3T^{2} \)
61 \( 1 + 12.9iT - 3.72e3T^{2} \)
67 \( 1 + (56.3 - 56.3i)T - 4.48e3iT^{2} \)
71 \( 1 + 92.7iT - 5.04e3T^{2} \)
73 \( 1 + (-95.4 + 95.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 100. iT - 6.24e3T^{2} \)
83 \( 1 + (5.62 + 5.62i)T + 6.88e3iT^{2} \)
89 \( 1 - 158. iT - 7.92e3T^{2} \)
97 \( 1 + (37.1 + 37.1i)T + 9.40e3iT^{2} \)
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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

−13.77262140663627879987305600620, −12.09348712048292026840380435044, −11.02755002729796228911862702954, −10.39527508841324698387080266171, −9.375472595004189910510440930486, −8.382236658189556849253852072775, −7.29058226658591029604868413036, −5.16829208547516700250827565740, −3.28018073878603964144738963872, −2.03451511104998771911319784382, 1.28441409378062444644494788939, 3.80004238183120353878832100283, 5.70181434744947773107074882990, 7.33925497631310564993669323187, 8.039811405883521225685586396255, 8.784509847600853825377851488446, 9.691427625491853899718195968763, 11.49267494848932948448548708571, 12.67903690134871575157292181981, 13.52260243654036982978360035886

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