Properties

Label 2-75-25.21-c3-0-1
Degree $2$
Conductor $75$
Sign $-0.421 - 0.906i$
Analytic cond. $4.42514$
Root an. cond. $2.10360$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.177 + 0.546i)2-s + (−2.42 − 1.76i)3-s + (6.20 + 4.50i)4-s + (−9.45 + 5.96i)5-s + (1.39 − 1.01i)6-s − 2.67·7-s + (−7.28 + 5.29i)8-s + (2.78 + 8.55i)9-s + (−1.58 − 6.22i)10-s + (−19.4 + 59.9i)11-s + (−7.11 − 21.8i)12-s + (4.38 + 13.4i)13-s + (0.475 − 1.46i)14-s + (33.4 + 2.17i)15-s + (17.3 + 53.4i)16-s + (−24.5 + 17.8i)17-s + ⋯
L(s)  = 1  + (−0.0627 + 0.193i)2-s + (−0.467 − 0.339i)3-s + (0.775 + 0.563i)4-s + (−0.845 + 0.533i)5-s + (0.0948 − 0.0689i)6-s − 0.144·7-s + (−0.321 + 0.233i)8-s + (0.103 + 0.317i)9-s + (−0.0500 − 0.196i)10-s + (−0.533 + 1.64i)11-s + (−0.171 − 0.526i)12-s + (0.0934 + 0.287i)13-s + (0.00907 − 0.0279i)14-s + (0.576 + 0.0375i)15-s + (0.271 + 0.834i)16-s + (−0.350 + 0.254i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.421 - 0.906i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ -0.421 - 0.906i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.522305 + 0.819031i\)
\(L(\frac12)\) \(\approx\) \(0.522305 + 0.819031i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.42 + 1.76i)T \)
5 \( 1 + (9.45 - 5.96i)T \)
good2 \( 1 + (0.177 - 0.546i)T + (-6.47 - 4.70i)T^{2} \)
7 \( 1 + 2.67T + 343T^{2} \)
11 \( 1 + (19.4 - 59.9i)T + (-1.07e3 - 782. i)T^{2} \)
13 \( 1 + (-4.38 - 13.4i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (24.5 - 17.8i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (22.2 - 16.1i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + (-35.9 + 110. i)T + (-9.84e3 - 7.15e3i)T^{2} \)
29 \( 1 + (-32.5 - 23.6i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (-180. + 130. i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-25.6 - 78.8i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-44.5 - 137. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 - 433.T + 7.95e4T^{2} \)
47 \( 1 + (371. + 269. i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (-200. - 145. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (-95.9 - 295. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (-0.644 + 1.98i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + (529. - 384. i)T + (9.29e4 - 2.86e5i)T^{2} \)
71 \( 1 + (-734. - 533. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (271. - 835. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (-921. - 669. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-798. + 579. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + (-305. + 940. i)T + (-5.70e5 - 4.14e5i)T^{2} \)
97 \( 1 + (1.24e3 + 903. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−14.74399129982279736062257214754, −12.91608812612285211062514743497, −12.18204826779229859627832530559, −11.25488528968690425144285334736, −10.20115411660845400319423002247, −8.236809456626495841483583200809, −7.25009353693080861285856963465, −6.45602272001462839506818925939, −4.38054459450318714933046093310, −2.47549548292755951919606840742, 0.65576484765097062151881614131, 3.26390744633930982015708149068, 5.13356824402742212494379342570, 6.33205086546918266119136344119, 7.88591188495519402772697815716, 9.238144412070354470700245639402, 10.74344475746156318953495214615, 11.26985819377199149633484259380, 12.31430189287760818441177768137, 13.63551218744715605184008863986

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