Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.444 + 0.896i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.686 − 2.56i)2-s + (−0.448 + 1.67i)3-s + (−2.62 + 1.51i)4-s + (3.50 − 3.56i)5-s + 4.59·6-s + (6.99 − 0.158i)7-s + (−1.82 − 1.82i)8-s + (−2.59 − 1.50i)9-s + (−11.5 − 6.54i)10-s + (−8.05 − 13.9i)11-s + (−1.35 − 5.06i)12-s + (−0.148 − 0.148i)13-s + (−5.20 − 17.8i)14-s + (4.38 + 7.46i)15-s + (−9.47 + 16.4i)16-s + (19.4 + 5.20i)17-s + ⋯
L(s)  = 1  + (−0.343 − 1.28i)2-s + (−0.149 + 0.557i)3-s + (−0.655 + 0.378i)4-s + (0.701 − 0.712i)5-s + 0.765·6-s + (0.999 − 0.0227i)7-s + (−0.227 − 0.227i)8-s + (−0.288 − 0.166i)9-s + (−1.15 − 0.654i)10-s + (−0.732 − 1.26i)11-s + (−0.113 − 0.422i)12-s + (−0.0114 − 0.0114i)13-s + (−0.372 − 1.27i)14-s + (0.292 + 0.497i)15-s + (−0.592 + 1.02i)16-s + (1.14 + 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.444 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.444 + 0.896i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (58, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ -0.444 + 0.896i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.647417 - 1.04338i\)
\(L(\frac12)\)  \(\approx\)  \(0.647417 - 1.04338i\)
\(L(2)\)   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 \( 1 + (0.448 - 1.67i)T \)
5 \( 1 + (-3.50 + 3.56i)T \)
7 \( 1 + (-6.99 + 0.158i)T \)
good2 \( 1 + (0.686 + 2.56i)T + (-3.46 + 2i)T^{2} \)
11 \( 1 + (8.05 + 13.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (0.148 + 0.148i)T + 169iT^{2} \)
17 \( 1 + (-19.4 - 5.20i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (10.9 + 6.34i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-15.6 + 4.18i)T + (458. - 264.5i)T^{2} \)
29 \( 1 - 21.7iT - 841T^{2} \)
31 \( 1 + (-7.17 - 12.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-16.8 - 62.8i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 2.26T + 1.68e3T^{2} \)
43 \( 1 + (-35.1 - 35.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (14.9 + 55.8i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (27.1 - 101. i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (59.5 - 34.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-17.3 + 30.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (31.9 + 8.56i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 59.5T + 5.04e3T^{2} \)
73 \( 1 + (8.96 - 33.4i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (2.87 + 1.65i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-4.48 - 4.48i)T + 6.88e3iT^{2} \)
89 \( 1 + (-123. - 71.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-113. + 113. i)T - 9.40e3iT^{2} \)
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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

−12.93881012751112068290193536306, −11.90140230789065426539654541681, −10.88663550451792542406736335552, −10.29282693261705256248718185081, −9.071552646935769436662398080414, −8.254740331124561712551666466603, −5.92749452464857988594495572857, −4.72426142747863171236241342644, −2.96800183754657509191865516808, −1.17796678169675625852217477559, 2.25919385517197929172538821417, 5.09091094879464166572020114720, 6.09453135557180163819370721632, 7.36877847970167963963075186577, 7.81507243225100928372173528534, 9.326283880007544452397858275550, 10.56202705109331964214575532717, 11.79202990867916699953837600832, 13.04013381608680127966231740567, 14.39596573261549603744886867676

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