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Properties

Label 2-375-15.2-c1-0-14
Degree $2$
Conductor $375$
Sign $0.193 - 0.981i$
Analytic cond. $2.99439$
Root an. cond. $1.73043$
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  + (−0.413 + 0.413i)2-s + (1.69 + 0.334i)3-s + 1.65i·4-s + (−0.841 + 0.564i)6-s + (2.93 + 2.93i)7-s + (−1.51 − 1.51i)8-s + (2.77 + 1.13i)9-s − 5.05i·11-s + (−0.555 + 2.81i)12-s + (−0.394 + 0.394i)13-s − 2.42·14-s − 2.06·16-s + (−4.21 + 4.21i)17-s + (−1.61 + 0.677i)18-s − 3.72i·19-s + ⋯
L(s)  = 1  + (−0.292 + 0.292i)2-s + (0.981 + 0.193i)3-s + 0.828i·4-s + (−0.343 + 0.230i)6-s + (1.10 + 1.10i)7-s + (−0.534 − 0.534i)8-s + (0.925 + 0.379i)9-s − 1.52i·11-s + (−0.160 + 0.813i)12-s + (−0.109 + 0.109i)13-s − 0.648·14-s − 0.516·16-s + (−1.02 + 1.02i)17-s + (−0.381 + 0.159i)18-s − 0.853i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.193 - 0.981i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.193 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31439 + 1.08069i\)
\(L(\frac12)\) \(\approx\) \(1.31439 + 1.08069i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 - 0.334i)T \)
5 \( 1 \)
good2 \( 1 + (0.413 - 0.413i)T - 2iT^{2} \)
7 \( 1 + (-2.93 - 2.93i)T + 7iT^{2} \)
11 \( 1 + 5.05iT - 11T^{2} \)
13 \( 1 + (0.394 - 0.394i)T - 13iT^{2} \)
17 \( 1 + (4.21 - 4.21i)T - 17iT^{2} \)
19 \( 1 + 3.72iT - 19T^{2} \)
23 \( 1 + (-1.85 - 1.85i)T + 23iT^{2} \)
29 \( 1 + 2.32T + 29T^{2} \)
31 \( 1 - 0.707T + 31T^{2} \)
37 \( 1 + (2.33 + 2.33i)T + 37iT^{2} \)
41 \( 1 + 4.45iT - 41T^{2} \)
43 \( 1 + (-4.74 + 4.74i)T - 43iT^{2} \)
47 \( 1 + (2.25 - 2.25i)T - 47iT^{2} \)
53 \( 1 + (-5.26 - 5.26i)T + 53iT^{2} \)
59 \( 1 + 0.326T + 59T^{2} \)
61 \( 1 - 1.14T + 61T^{2} \)
67 \( 1 + (7.14 + 7.14i)T + 67iT^{2} \)
71 \( 1 + 3.12iT - 71T^{2} \)
73 \( 1 + (-1.44 + 1.44i)T - 73iT^{2} \)
79 \( 1 + 0.0493iT - 79T^{2} \)
83 \( 1 + (8.20 + 8.20i)T + 83iT^{2} \)
89 \( 1 - 5.75T + 89T^{2} \)
97 \( 1 + (9.64 + 9.64i)T + 97iT^{2} \)
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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

−11.48789853433386983333765668631, −10.78258386160570484625670045818, −9.080036115350083853115148743146, −8.791982439135505175908735613386, −8.161958295747631665519589495442, −7.18573317184264561594272694711, −5.84705668743482855803961548808, −4.47527858803457122381376122038, −3.29108366885527087088589002123, −2.17626199266955888762955690157, 1.35587955473997280836853879302, 2.37828546418183980361270589353, 4.23478114394171542456029135204, 4.95987541729287215678631949593, 6.76752236479555250926805225373, 7.46545976977603112624248878272, 8.451126380972906077927746962364, 9.512013860283179468577235214533, 10.11721482339164072882494827838, 10.97468685464997222411661061279

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