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Properties

Label 2-105-35.34-c2-0-8
Degree $2$
Conductor $105$
Sign $0.390 - 0.920i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30i·2-s + 1.73·3-s − 1.29·4-s + (3.65 − 3.40i)5-s + 3.98i·6-s + (6.39 − 2.84i)7-s + 6.22i·8-s + 2.99·9-s + (7.84 + 8.41i)10-s − 13.9·11-s − 2.24·12-s − 3.78·13-s + (6.55 + 14.7i)14-s + (6.33 − 5.90i)15-s − 19.5·16-s − 14.8·17-s + ⋯
L(s)  = 1  + 1.15i·2-s + 0.577·3-s − 0.323·4-s + (0.731 − 0.681i)5-s + 0.664i·6-s + (0.913 − 0.406i)7-s + 0.778i·8-s + 0.333·9-s + (0.784 + 0.841i)10-s − 1.26·11-s − 0.186·12-s − 0.291·13-s + (0.467 + 1.05i)14-s + (0.422 − 0.393i)15-s − 1.21·16-s − 0.873·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.390 - 0.920i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.390 - 0.920i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.55334 + 1.02786i\)
\(L(\frac12)\) \(\approx\) \(1.55334 + 1.02786i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
5 \( 1 + (-3.65 + 3.40i)T \)
7 \( 1 + (-6.39 + 2.84i)T \)
good2 \( 1 - 2.30iT - 4T^{2} \)
11 \( 1 + 13.9T + 121T^{2} \)
13 \( 1 + 3.78T + 169T^{2} \)
17 \( 1 + 14.8T + 289T^{2} \)
19 \( 1 + 6.94iT - 361T^{2} \)
23 \( 1 - 40.2iT - 529T^{2} \)
29 \( 1 - 9.88T + 841T^{2} \)
31 \( 1 + 34.7iT - 961T^{2} \)
37 \( 1 - 30.4iT - 1.36e3T^{2} \)
41 \( 1 + 44.6iT - 1.68e3T^{2} \)
43 \( 1 + 26.0iT - 1.84e3T^{2} \)
47 \( 1 + 23.7T + 2.20e3T^{2} \)
53 \( 1 + 59.5iT - 2.80e3T^{2} \)
59 \( 1 - 81.0iT - 3.48e3T^{2} \)
61 \( 1 - 78.8iT - 3.72e3T^{2} \)
67 \( 1 - 29.9iT - 4.48e3T^{2} \)
71 \( 1 + 117.T + 5.04e3T^{2} \)
73 \( 1 + 22.2T + 5.32e3T^{2} \)
79 \( 1 - 142.T + 6.24e3T^{2} \)
83 \( 1 - 160.T + 6.88e3T^{2} \)
89 \( 1 + 66.3iT - 7.92e3T^{2} \)
97 \( 1 + 45.1T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.66438572406845554869272097156, −13.35618720312574657200281214031, −11.62506842759301844745432083766, −10.34314511264051949669598164666, −8.998928942942964738786680460684, −8.061034348149022130455000132788, −7.22010054044756659649160780914, −5.63987285911613251103258578516, −4.71811413908102375492259938488, −2.16712314248157374089833871286, 2.03261663455214640108800298541, 2.88035497817553577995021160220, 4.78220900764237866444806786337, 6.55914332901292398803294095913, 7.998486303787939974429794114100, 9.288231729297035357171037953089, 10.48466402809816483253404481271, 10.91090188605760209724043587369, 12.29488742036739502735321256765, 13.16824411526176283996228696106

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