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Properties

Label 2-105-3.2-c2-0-7
Degree $2$
Conductor $105$
Sign $0.234 - 0.972i$
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  + 2.02i·2-s + (2.91 + 0.703i)3-s − 0.0906·4-s − 2.23i·5-s + (−1.42 + 5.89i)6-s + 2.64·7-s + 7.90i·8-s + (8.00 + 4.10i)9-s + 4.52·10-s − 11.6i·11-s + (−0.264 − 0.0638i)12-s − 20.3·13-s + 5.35i·14-s + (1.57 − 6.52i)15-s − 16.3·16-s + 9.92i·17-s + ⋯
L(s)  = 1  + 1.01i·2-s + (0.972 + 0.234i)3-s − 0.0226·4-s − 0.447i·5-s + (−0.237 + 0.983i)6-s + 0.377·7-s + 0.988i·8-s + (0.889 + 0.456i)9-s + 0.452·10-s − 1.05i·11-s + (−0.0220 − 0.00531i)12-s − 1.56·13-s + 0.382i·14-s + (0.104 − 0.434i)15-s − 1.02·16-s + 0.583i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 - 0.972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.234 - 0.972i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.234 - 0.972i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.51619 + 1.19378i\)
\(L(\frac12)\) \(\approx\) \(1.51619 + 1.19378i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.91 - 0.703i)T \)
5 \( 1 + 2.23iT \)
7 \( 1 - 2.64T \)
good2 \( 1 - 2.02iT - 4T^{2} \)
11 \( 1 + 11.6iT - 121T^{2} \)
13 \( 1 + 20.3T + 169T^{2} \)
17 \( 1 - 9.92iT - 289T^{2} \)
19 \( 1 + 4.91T + 361T^{2} \)
23 \( 1 + 11.4iT - 529T^{2} \)
29 \( 1 + 37.5iT - 841T^{2} \)
31 \( 1 + 21.2T + 961T^{2} \)
37 \( 1 - 57.1T + 1.36e3T^{2} \)
41 \( 1 + 20.2iT - 1.68e3T^{2} \)
43 \( 1 - 5.88T + 1.84e3T^{2} \)
47 \( 1 + 74.5iT - 2.20e3T^{2} \)
53 \( 1 - 88.2iT - 2.80e3T^{2} \)
59 \( 1 - 71.6iT - 3.48e3T^{2} \)
61 \( 1 + 26.0T + 3.72e3T^{2} \)
67 \( 1 - 76.4T + 4.48e3T^{2} \)
71 \( 1 - 47.1iT - 5.04e3T^{2} \)
73 \( 1 + 135.T + 5.32e3T^{2} \)
79 \( 1 + 98.2T + 6.24e3T^{2} \)
83 \( 1 - 55.8iT - 6.88e3T^{2} \)
89 \( 1 + 105. iT - 7.92e3T^{2} \)
97 \( 1 - 61.8T + 9.40e3T^{2} \)
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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

−14.14411295092885525892303670537, −13.04058600393880381381379121035, −11.70206286056763408723245644864, −10.35746220801586425516558142025, −9.009101114702077251216220095186, −8.143300702741756179230823436525, −7.31367001537764195171708602196, −5.78011209253464086306401167049, −4.43181936005075208988467504123, −2.43697135777180525035688670486, 1.91547392248098776952418801719, 2.98517788854690293632259279248, 4.54900646445797926594283027569, 6.93450601132634560130308246034, 7.63699362538394923762577640549, 9.408036677034749781317706577958, 9.988141761055892934224539900066, 11.24946460872778631242064269304, 12.34054267970465487232478422405, 12.98997082514759461237881186068

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