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Properties

Label 2-105-35.24-c2-0-7
Degree $2$
Conductor $105$
Sign $0.629 + 0.777i$
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  + (−2.95 + 1.70i)2-s + (−0.866 + 1.5i)3-s + (3.80 − 6.59i)4-s + (−3.57 + 3.49i)5-s − 5.90i·6-s + (−5.14 − 4.74i)7-s + 12.3i·8-s + (−1.5 − 2.59i)9-s + (4.58 − 16.4i)10-s + (0.694 − 1.20i)11-s + (6.59 + 11.4i)12-s + 3.78·13-s + (23.2 + 5.22i)14-s + (−2.15 − 8.38i)15-s + (−5.76 − 9.98i)16-s + (15.8 − 27.4i)17-s + ⋯
L(s)  = 1  + (−1.47 + 0.852i)2-s + (−0.288 + 0.5i)3-s + (0.951 − 1.64i)4-s + (−0.714 + 0.699i)5-s − 0.983i·6-s + (−0.735 − 0.677i)7-s + 1.54i·8-s + (−0.166 − 0.288i)9-s + (0.458 − 1.64i)10-s + (0.0631 − 0.109i)11-s + (0.549 + 0.951i)12-s + 0.291·13-s + (1.66 + 0.373i)14-s + (−0.143 − 0.559i)15-s + (−0.360 − 0.624i)16-s + (0.932 − 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.629 + 0.777i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.629 + 0.777i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.198834 - 0.0948139i\)
\(L(\frac12)\) \(\approx\) \(0.198834 - 0.0948139i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 1.5i)T \)
5 \( 1 + (3.57 - 3.49i)T \)
7 \( 1 + (5.14 + 4.74i)T \)
good2 \( 1 + (2.95 - 1.70i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (-0.694 + 1.20i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 3.78T + 169T^{2} \)
17 \( 1 + (-15.8 + 27.4i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-12.8 + 7.40i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (19.5 - 11.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 42.9T + 841T^{2} \)
31 \( 1 + (28.0 + 16.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (7.29 - 4.21i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 46.3iT - 1.68e3T^{2} \)
43 \( 1 + 9.92iT - 1.84e3T^{2} \)
47 \( 1 + (27.2 + 47.2i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-74.4 - 42.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (80.4 + 46.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-1.41 + 0.819i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (37.9 + 21.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 60.9T + 5.04e3T^{2} \)
73 \( 1 + (32.3 - 55.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-0.744 - 1.29i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 13.6T + 6.88e3T^{2} \)
89 \( 1 + (-110. + 63.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 119.T + 9.40e3T^{2} \)
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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

−13.73774543976919064148473440553, −11.83358576701739311200357298470, −10.85596037783837319833572604759, −9.937798285633835008421042849496, −9.170671467798486093639855808091, −7.60219820801032514175198991706, −7.09036821956956015814322194775, −5.72625865629082366216568942242, −3.57249619785463407416659789741, −0.28098425709891557799081140645, 1.55150528646435791841103484445, 3.49272498635186478734361680951, 5.85221001768809212245233723735, 7.54733285939515276427833796576, 8.376870029524627764008703106867, 9.328779169643587338128618755072, 10.42071919530366302588632869723, 11.58621683816567722668777846219, 12.33034225650256840605990112522, 12.94152336957977292953985244592

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