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Properties

Label 2-75-15.2-c1-0-0
Degree $2$
Conductor $75$
Sign $0.326 - 0.945i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 1.22i)3-s + 2i·4-s + (1.22 + 1.22i)7-s − 2.99i·9-s + (−2.44 − 2.44i)12-s + (3.67 − 3.67i)13-s − 4·16-s − i·19-s − 2.99·21-s + (3.67 + 3.67i)27-s + (−2.44 + 2.44i)28-s + 7·31-s + 5.99·36-s + (−4.89 − 4.89i)37-s + 9i·39-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + i·4-s + (0.462 + 0.462i)7-s − 0.999i·9-s + (−0.707 − 0.707i)12-s + (1.01 − 1.01i)13-s − 16-s − 0.229i·19-s − 0.654·21-s + (0.707 + 0.707i)27-s + (−0.462 + 0.462i)28-s + 1.25·31-s + 0.999·36-s + (−0.805 − 0.805i)37-s + 1.44i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.326 - 0.945i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1/2),\ 0.326 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652322 + 0.464859i\)
\(L(\frac12)\) \(\approx\) \(0.652322 + 0.464859i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
good2 \( 1 - 2iT^{2} \)
7 \( 1 + (-1.22 - 1.22i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-3.67 + 3.67i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + iT - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 7T + 31T^{2} \)
37 \( 1 + (4.89 + 4.89i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (8.57 - 8.57i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + (11.0 + 11.0i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-9.79 + 9.79i)T - 73iT^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-13.4 - 13.4i)T + 97iT^{2} \)
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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

−15.10711881946085356342201389570, −13.52130204400553857303362613092, −12.38381364700783856645901131780, −11.51549491059053192599886655942, −10.53054949609946400449155059232, −9.036581819683474654658054851719, −7.994397841346464287665603950071, −6.30718475875464681992262523024, −4.86441124038730884874123218676, −3.36245573472265835272861328486, 1.51080945135387386955239285171, 4.64167692735357968724799275556, 6.01022959873947222877292291735, 6.98889702694688790200163282196, 8.550953013005702770550571426653, 10.15975893520387179183476691193, 11.08796799855110413220929655907, 11.94290278012256817197617092404, 13.54356823819002703178941061605, 14.01623375393655953233815972793

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