Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.642 + 0.766i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.63 − 2.63i)2-s + (−1.59 − 2.54i)3-s − 9.88i·4-s + (4.15 + 2.78i)5-s + (−10.8 − 2.49i)6-s + (2.39 + 6.57i)7-s + (−15.5 − 15.5i)8-s + (−3.91 + 8.10i)9-s + (18.2 − 3.60i)10-s + 5.66i·11-s + (−25.1 + 15.7i)12-s + (−1.68 − 1.68i)13-s + (23.6 + 11.0i)14-s + (0.449 − 14.9i)15-s − 42.1·16-s + (−14.0 − 14.0i)17-s + ⋯
L(s)  = 1  + (1.31 − 1.31i)2-s + (−0.531 − 0.846i)3-s − 2.47i·4-s + (0.830 + 0.556i)5-s + (−1.81 − 0.415i)6-s + (0.342 + 0.939i)7-s + (−1.93 − 1.93i)8-s + (−0.434 + 0.900i)9-s + (1.82 − 0.360i)10-s + 0.514i·11-s + (−2.09 + 1.31i)12-s + (−0.129 − 0.129i)13-s + (1.68 + 0.786i)14-s + (0.0299 − 0.999i)15-s − 2.63·16-s + (−0.824 − 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.642 + 0.766i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (62, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.642 + 0.766i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.00277 - 2.14867i\)
\(L(\frac12)\)  \(\approx\)  \(1.00277 - 2.14867i\)
\(L(2)\)   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 \( 1 + (1.59 + 2.54i)T \)
5 \( 1 + (-4.15 - 2.78i)T \)
7 \( 1 + (-2.39 - 6.57i)T \)
good2 \( 1 + (-2.63 + 2.63i)T - 4iT^{2} \)
11 \( 1 - 5.66iT - 121T^{2} \)
13 \( 1 + (1.68 + 1.68i)T + 169iT^{2} \)
17 \( 1 + (14.0 + 14.0i)T + 289iT^{2} \)
19 \( 1 - 24.0T + 361T^{2} \)
23 \( 1 + (3.17 + 3.17i)T + 529iT^{2} \)
29 \( 1 + 24.1T + 841T^{2} \)
31 \( 1 - 23.8iT - 961T^{2} \)
37 \( 1 + (13.8 + 13.8i)T + 1.36e3iT^{2} \)
41 \( 1 + 53.4T + 1.68e3T^{2} \)
43 \( 1 + (-25.9 + 25.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (-27.3 - 27.3i)T + 2.20e3iT^{2} \)
53 \( 1 + (22.4 + 22.4i)T + 2.80e3iT^{2} \)
59 \( 1 + 14.2iT - 3.48e3T^{2} \)
61 \( 1 - 90.2iT - 3.72e3T^{2} \)
67 \( 1 + (0.492 + 0.492i)T + 4.48e3iT^{2} \)
71 \( 1 - 54.2iT - 5.04e3T^{2} \)
73 \( 1 + (-30.4 - 30.4i)T + 5.32e3iT^{2} \)
79 \( 1 + 58.3iT - 6.24e3T^{2} \)
83 \( 1 + (-55.7 + 55.7i)T - 6.88e3iT^{2} \)
89 \( 1 + 109. iT - 7.92e3T^{2} \)
97 \( 1 + (-48.0 + 48.0i)T - 9.40e3iT^{2} \)
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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

−13.04670738566789600288681143465, −12.09427626237664263195390452776, −11.44291717954434896489501351067, −10.46657640854074865051659454979, −9.274599735477004890980496823460, −7.01044414171779251455157635305, −5.74728499447465013546632251002, −5.03190195911048051555170222681, −2.80346253118575829556353297739, −1.78444511111767206639241343531, 3.70136218730463718310653099938, 4.78439285098652701533976247251, 5.67585341026506077826065798674, 6.69732229254400685432369777097, 8.149496145221109222445279102982, 9.459343414367113294818286839770, 10.93027056126484746851521343286, 12.12531824980799748329397522993, 13.34048229980223780575607621048, 13.90748867818721802952118830277

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