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Properties

Label 2-105-35.34-c2-0-2
Degree $2$
Conductor $105$
Sign $-0.999 + 0.0249i$
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  + 3.80i·2-s + 1.73·3-s − 10.4·4-s + (3.71 + 3.34i)5-s + 6.59i·6-s + (−5.08 + 4.81i)7-s − 24.6i·8-s + 2.99·9-s + (−12.7 + 14.1i)10-s − 9.90·11-s − 18.1·12-s + 9.60·13-s + (−18.3 − 19.3i)14-s + (6.43 + 5.79i)15-s + 52.0·16-s + 28.8·17-s + ⋯
L(s)  = 1  + 1.90i·2-s + 0.577·3-s − 2.62·4-s + (0.742 + 0.669i)5-s + 1.09i·6-s + (−0.725 + 0.687i)7-s − 3.08i·8-s + 0.333·9-s + (−1.27 + 1.41i)10-s − 0.900·11-s − 1.51·12-s + 0.738·13-s + (−1.30 − 1.38i)14-s + (0.428 + 0.386i)15-s + 3.25·16-s + 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0249i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.999 + 0.0249i$
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: \((2,\ 105,\ (\ :1),\ -0.999 + 0.0249i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0172437 - 1.38406i\)
\(L(\frac12)\) \(\approx\) \(0.0172437 - 1.38406i\)
\(L(2)\) 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 \( 1 + (-3.71 - 3.34i)T \)
7 \( 1 + (5.08 - 4.81i)T \)
good2 \( 1 - 3.80iT - 4T^{2} \)
11 \( 1 + 9.90T + 121T^{2} \)
13 \( 1 - 9.60T + 169T^{2} \)
17 \( 1 - 28.8T + 289T^{2} \)
19 \( 1 + 2.29iT - 361T^{2} \)
23 \( 1 - 1.25iT - 529T^{2} \)
29 \( 1 - 2.52T + 841T^{2} \)
31 \( 1 - 8.89iT - 961T^{2} \)
37 \( 1 - 23.8iT - 1.36e3T^{2} \)
41 \( 1 + 59.7iT - 1.68e3T^{2} \)
43 \( 1 - 42.1iT - 1.84e3T^{2} \)
47 \( 1 - 41.8T + 2.20e3T^{2} \)
53 \( 1 + 47.0iT - 2.80e3T^{2} \)
59 \( 1 + 19.2iT - 3.48e3T^{2} \)
61 \( 1 - 12.9iT - 3.72e3T^{2} \)
67 \( 1 + 120. iT - 4.48e3T^{2} \)
71 \( 1 - 36.1T + 5.04e3T^{2} \)
73 \( 1 + 61.8T + 5.32e3T^{2} \)
79 \( 1 + 45.8T + 6.24e3T^{2} \)
83 \( 1 - 22.9T + 6.88e3T^{2} \)
89 \( 1 + 88.8iT - 7.92e3T^{2} \)
97 \( 1 - 18.4T + 9.40e3T^{2} \)
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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

−14.23712553528655593141361926291, −13.52002190835425199299687755230, −12.62402451743025360104958987058, −10.24300974193504968423049590282, −9.414757267824226061796559011314, −8.360659410462170524204123325528, −7.30610108686771594618061537172, −6.17291376783719856447299604108, −5.35667435339372034056520571749, −3.30000124273683759830154124239, 1.11976874397297022956802521971, 2.79714857969737703114227001678, 4.01922844828609481361062592490, 5.52964525898652803699938860999, 7.995818460698903430494779227904, 9.152922312158551729997954539819, 10.00622725331156271460295443950, 10.59661290872874955298889201221, 12.13167246236109961865381915927, 12.99235799527972421332297487788

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