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Properties

Label 2-105-105.74-c2-0-7
Degree $2$
Conductor $105$
Sign $-0.828 - 0.559i$
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  + (1.48 + 2.57i)2-s + (−1.18 + 2.75i)3-s + (−2.43 + 4.22i)4-s + (4.91 − 0.901i)5-s + (−8.87 + 1.06i)6-s + (−5.16 + 4.72i)7-s − 2.60·8-s + (−6.21 − 6.51i)9-s + (9.64 + 11.3i)10-s + (3.56 + 2.05i)11-s + (−8.76 − 11.7i)12-s − 21.3i·13-s + (−19.8 − 6.30i)14-s + (−3.32 + 14.6i)15-s + (5.87 + 10.1i)16-s + (1.11 − 1.92i)17-s + ⋯
L(s)  = 1  + (0.744 + 1.28i)2-s + (−0.393 + 0.919i)3-s + (−0.609 + 1.05i)4-s + (0.983 − 0.180i)5-s + (−1.47 + 0.176i)6-s + (−0.738 + 0.674i)7-s − 0.325·8-s + (−0.690 − 0.723i)9-s + (0.964 + 1.13i)10-s + (0.323 + 0.186i)11-s + (−0.730 − 0.975i)12-s − 1.64i·13-s + (−1.41 − 0.450i)14-s + (−0.221 + 0.975i)15-s + (0.367 + 0.635i)16-s + (0.0653 − 0.113i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 - 0.559i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.828 - 0.559i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.828 - 0.559i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.540085 + 1.76561i\)
\(L(\frac12)\) \(\approx\) \(0.540085 + 1.76561i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.18 - 2.75i)T \)
5 \( 1 + (-4.91 + 0.901i)T \)
7 \( 1 + (5.16 - 4.72i)T \)
good2 \( 1 + (-1.48 - 2.57i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (-3.56 - 2.05i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 21.3iT - 169T^{2} \)
17 \( 1 + (-1.11 + 1.92i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-4.93 - 8.55i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-9.66 - 16.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 28.1iT - 841T^{2} \)
31 \( 1 + (-17.6 + 30.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (43.1 - 24.9i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 27.4iT - 1.68e3T^{2} \)
43 \( 1 + 41.9iT - 1.84e3T^{2} \)
47 \( 1 + (19.1 + 33.2i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-25.6 + 44.4i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (22.0 + 12.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (34.1 + 59.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (59.1 + 34.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 11.9iT - 5.04e3T^{2} \)
73 \( 1 + (-88.6 - 51.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-32.1 - 55.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 122.T + 6.88e3T^{2} \)
89 \( 1 + (116. - 67.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 13.6iT - 9.40e3T^{2} \)
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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

−14.17018405571802995392582491277, −13.15224556485278910839330784816, −12.24359348529869502530921793816, −10.51193130920715806099691352197, −9.631606402218720516539494485603, −8.454066324349724163971269599172, −6.74021071239464131260439894144, −5.65277560934353916903710308901, −5.21589323808236746022929174105, −3.39343065907656789796931377177, 1.41629332604352798620029223162, 2.81785623249694407595296026734, 4.56648693633757129722320938162, 6.16631111493728511359034507638, 7.08033293609686930239329640554, 9.148394052084886835809520833454, 10.29636145045679541905188775361, 11.20714950510746589802439614342, 12.15707085791459067347999284560, 13.06217815851365842833293305018

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