Properties

Label 2-105-35.24-c2-0-5
Degree $2$
Conductor $105$
Sign $0.832 - 0.553i$
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  + (−2.53 + 1.46i)2-s + (−0.866 + 1.5i)3-s + (2.29 − 3.98i)4-s + (1.01 − 4.89i)5-s − 5.07i·6-s + (6.93 − 0.976i)7-s + 1.75i·8-s + (−1.5 − 2.59i)9-s + (4.61 + 13.9i)10-s + (−2.31 + 4.00i)11-s + (3.98 + 6.89i)12-s + 14.9·13-s + (−16.1 + 12.6i)14-s + (6.46 + 5.75i)15-s + (6.62 + 11.4i)16-s + (−8.38 + 14.5i)17-s + ⋯
L(s)  = 1  + (−1.26 + 0.733i)2-s + (−0.288 + 0.5i)3-s + (0.574 − 0.995i)4-s + (0.202 − 0.979i)5-s − 0.846i·6-s + (0.990 − 0.139i)7-s + 0.219i·8-s + (−0.166 − 0.288i)9-s + (0.461 + 1.39i)10-s + (−0.210 + 0.363i)11-s + (0.331 + 0.574i)12-s + 1.15·13-s + (−1.15 + 0.903i)14-s + (0.431 + 0.383i)15-s + (0.414 + 0.717i)16-s + (−0.493 + 0.853i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.832 - 0.553i$
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: \((2,\ 105,\ (\ :1),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.719329 + 0.217285i\)
\(L(\frac12)\) \(\approx\) \(0.719329 + 0.217285i\)
\(L(2)\) 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 \( 1 + (-1.01 + 4.89i)T \)
7 \( 1 + (-6.93 + 0.976i)T \)
good2 \( 1 + (2.53 - 1.46i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (2.31 - 4.00i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 14.9T + 169T^{2} \)
17 \( 1 + (8.38 - 14.5i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-15.2 + 8.78i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-36.9 + 21.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 18.1T + 841T^{2} \)
31 \( 1 + (-3.46 - 2.00i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (24.5 - 14.1i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 44.5iT - 1.68e3T^{2} \)
43 \( 1 + 44.3iT - 1.84e3T^{2} \)
47 \( 1 + (-30.3 - 52.6i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-46.8 - 27.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-1.54 - 0.891i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (44.4 - 25.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.44 + 1.41i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 19.2T + 5.04e3T^{2} \)
73 \( 1 + (49.4 - 85.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (59.7 + 103. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 89.3T + 6.88e3T^{2} \)
89 \( 1 + (97.2 - 56.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 115.T + 9.40e3T^{2} \)
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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

−13.73219248009021040880851584009, −12.50853787372073339845001121563, −11.08153889098631505791761292182, −10.25341840979493094328702740714, −8.842009190151476158820888871262, −8.577055660285010637346966941896, −7.15218738015732680319694141307, −5.69694976206654802147215437778, −4.40462117609507848768047911667, −1.13931340815714377923464723980, 1.35654980830136719545829362664, 2.92847381745919312406017610232, 5.46979389194069433008639052999, 7.06106145002405298824030618098, 8.075865241530900657369278563092, 9.141838116215140922235608072313, 10.44094184784614230190836421524, 11.27907520444877825682653485207, 11.65278136924308220609346957727, 13.41648990595434171340271268056

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