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Properties

Label 2-75-5.4-c17-0-30
Degree $2$
Conductor $75$
Sign $0.894 - 0.447i$
Analytic cond. $137.416$
Root an. cond. $11.7224$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 367. i·2-s − 6.56e3i·3-s − 3.77e3·4-s + 2.40e6·6-s + 1.91e7i·7-s + 4.67e7i·8-s − 4.30e7·9-s − 1.08e9·11-s + 2.47e7i·12-s − 4.07e9i·13-s − 7.02e9·14-s − 1.76e10·16-s + 1.03e9i·17-s − 1.58e10i·18-s + 8.27e10·19-s + ⋯
L(s)  = 1  + 1.01i·2-s − 0.577i·3-s − 0.0288·4-s + 0.585·6-s + 1.25i·7-s + 0.985i·8-s − 0.333·9-s − 1.52·11-s + 0.0166i·12-s − 1.38i·13-s − 1.27·14-s − 1.02·16-s + 0.0359i·17-s − 0.338i·18-s + 1.11·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(137.416\)
Root analytic conductor: \(11.7224\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :17/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.853625467\)
\(L(\frac12)\) \(\approx\) \(1.853625467\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 6.56e3iT \)
5 \( 1 \)
good2 \( 1 - 367. iT - 1.31e5T^{2} \)
7 \( 1 - 1.91e7iT - 2.32e14T^{2} \)
11 \( 1 + 1.08e9T + 5.05e17T^{2} \)
13 \( 1 + 4.07e9iT - 8.65e18T^{2} \)
17 \( 1 - 1.03e9iT - 8.27e20T^{2} \)
19 \( 1 - 8.27e10T + 5.48e21T^{2} \)
23 \( 1 + 5.01e11iT - 1.41e23T^{2} \)
29 \( 1 + 3.86e12T + 7.25e24T^{2} \)
31 \( 1 - 8.19e12T + 2.25e25T^{2} \)
37 \( 1 + 1.23e13iT - 4.56e26T^{2} \)
41 \( 1 + 7.08e13T + 2.61e27T^{2} \)
43 \( 1 + 3.21e13iT - 5.87e27T^{2} \)
47 \( 1 + 1.79e13iT - 2.66e28T^{2} \)
53 \( 1 + 1.35e14iT - 2.05e29T^{2} \)
59 \( 1 - 1.26e15T + 1.27e30T^{2} \)
61 \( 1 - 2.30e15T + 2.24e30T^{2} \)
67 \( 1 - 4.62e15iT - 1.10e31T^{2} \)
71 \( 1 - 1.04e16T + 2.96e31T^{2} \)
73 \( 1 - 6.57e15iT - 4.74e31T^{2} \)
79 \( 1 + 7.36e15T + 1.81e32T^{2} \)
83 \( 1 + 6.77e15iT - 4.21e32T^{2} \)
89 \( 1 + 6.56e15T + 1.37e33T^{2} \)
97 \( 1 + 2.22e16iT - 5.95e33T^{2} \)
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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

−11.42346191026126742300392753793, −10.16043937104622222014631394378, −8.479155904301583288251405172228, −7.993509898294708350430005736092, −6.88588729885263950525869981339, −5.55548893961190702095443426511, −5.33546081143352208308075939690, −2.87572494090633726264944847592, −2.24872285872227428803530140194, −0.47616834394617611829573787102, 0.74954968257018501273065946316, 1.88538743438754045767936707652, 3.12991836878344683363255992524, 3.98572623434793826644174031359, 5.12803417062588703644224483172, 6.81039564621140868239436431180, 7.80724939178179303441352216272, 9.550348228551179674641408060760, 10.14341637481360951401743871306, 11.11337953875972668571660620019

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