Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.247 - 0.968i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.582 + 2.17i)2-s + (1.71 + 0.245i)3-s + (−2.64 + 1.52i)4-s + (−2.21 − 0.337i)5-s + (0.465 + 3.86i)6-s + (−1.15 − 2.38i)7-s + (−1.68 − 1.68i)8-s + (2.87 + 0.840i)9-s + (−0.552 − 4.99i)10-s + (3.88 − 2.24i)11-s + (−4.91 + 1.97i)12-s + (−1.08 + 1.08i)13-s + (4.50 − 3.88i)14-s + (−3.70 − 1.12i)15-s + (−0.381 + 0.660i)16-s + (−2.04 − 0.548i)17-s + ⋯
L(s)  = 1  + (0.411 + 1.53i)2-s + (0.989 + 0.141i)3-s + (−1.32 + 0.764i)4-s + (−0.988 − 0.151i)5-s + (0.189 + 1.57i)6-s + (−0.435 − 0.900i)7-s + (−0.595 − 0.595i)8-s + (0.959 + 0.280i)9-s + (−0.174 − 1.58i)10-s + (1.17 − 0.676i)11-s + (−1.41 + 0.569i)12-s + (−0.300 + 0.300i)13-s + (1.20 − 1.03i)14-s + (−0.957 − 0.289i)15-s + (−0.0952 + 0.165i)16-s + (−0.496 − 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.247 - 0.968i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.247 - 0.968i)$
$L(1)$  $\approx$  $0.840732 + 1.08202i$
$L(\frac12)$  $\approx$  $0.840732 + 1.08202i$
$L(\frac{3}{2})$   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 \( 1 + (-1.71 - 0.245i)T \)
5 \( 1 + (2.21 + 0.337i)T \)
7 \( 1 + (1.15 + 2.38i)T \)
good2 \( 1 + (-0.582 - 2.17i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-3.88 + 2.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.08 - 1.08i)T - 13iT^{2} \)
17 \( 1 + (2.04 + 0.548i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.66 + 2.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.13 - 0.840i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 + (-0.530 - 0.918i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.75 - 1.54i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.84iT - 41T^{2} \)
43 \( 1 + (-2.00 + 2.00i)T - 43iT^{2} \)
47 \( 1 + (-1.36 - 5.10i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.23 - 8.34i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.35 - 4.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.88 + 6.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.152 - 0.569i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.66iT - 71T^{2} \)
73 \( 1 + (4.22 + 1.13i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.78 - 3.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.0 - 11.0i)T + 83iT^{2} \)
89 \( 1 + (-1.75 + 3.04i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.60 + 5.60i)T + 97iT^{2} \)
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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

−14.15385407667447482908453352813, −13.65020947462888697327419471252, −12.42283570738667510988642423009, −10.84417210804533507238928597688, −9.177662431390855691022917616323, −8.365887542420915940563285512593, −7.30489529983975625283158222108, −6.56334201852208209089130407706, −4.50088022144951041798946409380, −3.73842716984026107285574706401, 2.13362566073489665501973051282, 3.46629878377257557528780203577, 4.41670698101307142749758913995, 6.79046811394826588938736089671, 8.351318739755326587414626483886, 9.354200011484800913347302549242, 10.33831293931580869426993309300, 11.73913619839738844938358843868, 12.32375679559964067762103632361, 13.06957107582583010493188919811

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