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Properties

Label 2-75-5.4-c17-0-22
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  + 702. i·2-s + 6.56e3i·3-s − 3.62e5·4-s − 4.60e6·6-s + 2.67e6i·7-s − 1.62e8i·8-s − 4.30e7·9-s − 9.86e8·11-s − 2.37e9i·12-s + 2.81e9i·13-s − 1.87e9·14-s + 6.66e10·16-s − 1.28e10i·17-s − 3.02e10i·18-s + 1.22e11·19-s + ⋯
L(s)  = 1  + 1.94i·2-s + 0.577i·3-s − 2.76·4-s − 1.12·6-s + 0.175i·7-s − 3.42i·8-s − 0.333·9-s − 1.38·11-s − 1.59i·12-s + 0.957i·13-s − 0.340·14-s + 3.87·16-s − 0.446i·17-s − 0.646i·18-s + 1.65·19-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

\(L(9)\) \(\approx\) \(1.528552836\)
\(L(\frac12)\) \(\approx\) \(1.528552836\)
\(L(\frac{19}{2})\) 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 \( 1 - 702. iT - 1.31e5T^{2} \)
7 \( 1 - 2.67e6iT - 2.32e14T^{2} \)
11 \( 1 + 9.86e8T + 5.05e17T^{2} \)
13 \( 1 - 2.81e9iT - 8.65e18T^{2} \)
17 \( 1 + 1.28e10iT - 8.27e20T^{2} \)
19 \( 1 - 1.22e11T + 5.48e21T^{2} \)
23 \( 1 - 1.14e11iT - 1.41e23T^{2} \)
29 \( 1 - 4.90e12T + 7.25e24T^{2} \)
31 \( 1 - 7.64e12T + 2.25e25T^{2} \)
37 \( 1 + 3.18e12iT - 4.56e26T^{2} \)
41 \( 1 + 5.53e12T + 2.61e27T^{2} \)
43 \( 1 - 2.42e13iT - 5.87e27T^{2} \)
47 \( 1 - 5.82e13iT - 2.66e28T^{2} \)
53 \( 1 + 6.45e14iT - 2.05e29T^{2} \)
59 \( 1 - 1.04e12T + 1.27e30T^{2} \)
61 \( 1 + 1.56e15T + 2.24e30T^{2} \)
67 \( 1 + 1.31e15iT - 1.10e31T^{2} \)
71 \( 1 + 7.09e15T + 2.96e31T^{2} \)
73 \( 1 + 1.17e16iT - 4.74e31T^{2} \)
79 \( 1 - 7.14e15T + 1.81e32T^{2} \)
83 \( 1 - 9.04e15iT - 4.21e32T^{2} \)
89 \( 1 - 1.16e16T + 1.37e33T^{2} \)
97 \( 1 - 8.32e16iT - 5.95e33T^{2} \)
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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

−12.01444646656648673068747647363, −10.19141911403977365999319371817, −9.346662055336586782469791734365, −8.294189089719489111861764496187, −7.41319365882501706429226012637, −6.27200065620536567375818759154, −5.17344317349882825359931123842, −4.57326410071302493317017254564, −3.07103670889386720000210344691, −0.69873428775041787930569473586, 0.54186718734903349866426691956, 1.19119118920844315047201689968, 2.61168557499666695458067421241, 3.07629128501760485231213282952, 4.61126877706638396075016004139, 5.61006862042028971602528961143, 7.74594123174033737876992078207, 8.587554147489120287040266163321, 10.06828512695803637058217285999, 10.52902967560278570370941233110

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