Properties

Label 2-375-25.16-c1-0-3
Degree $2$
Conductor $375$
Sign $0.344 + 0.938i$
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  + (−1.38 − 1.00i)2-s + (−0.309 − 0.951i)3-s + (0.281 + 0.867i)4-s + (−0.527 + 1.62i)6-s + 3.94·7-s + (−0.573 + 1.76i)8-s + (−0.809 + 0.587i)9-s + (4.78 + 3.47i)11-s + (0.737 − 0.535i)12-s + (2.66 − 1.93i)13-s + (−5.44 − 3.95i)14-s + (4.03 − 2.93i)16-s + (−0.836 + 2.57i)17-s + 1.70·18-s + (−0.728 + 2.24i)19-s + ⋯
L(s)  = 1  + (−0.976 − 0.709i)2-s + (−0.178 − 0.549i)3-s + (0.140 + 0.433i)4-s + (−0.215 + 0.662i)6-s + 1.49·7-s + (−0.202 + 0.624i)8-s + (−0.269 + 0.195i)9-s + (1.44 + 1.04i)11-s + (0.212 − 0.154i)12-s + (0.738 − 0.536i)13-s + (−1.45 − 1.05i)14-s + (1.00 − 0.733i)16-s + (−0.202 + 0.624i)17-s + 0.402·18-s + (−0.167 + 0.514i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.344 + 0.938i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.344 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.772259 - 0.539240i\)
\(L(\frac12)\) \(\approx\) \(0.772259 - 0.539240i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 \)
good2 \( 1 + (1.38 + 1.00i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 - 3.94T + 7T^{2} \)
11 \( 1 + (-4.78 - 3.47i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.66 + 1.93i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.836 - 2.57i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.728 - 2.24i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.472 + 0.343i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.20 + 3.72i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.837 + 2.57i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.0168 - 0.0122i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.19 - 0.865i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.27T + 43T^{2} \)
47 \( 1 + (1.67 + 5.16i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.870 + 2.67i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.79 + 2.75i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.51 + 3.28i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (1.86 - 5.73i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.50 + 7.70i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-10.7 - 7.82i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-5.14 - 15.8i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.241 - 0.743i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-2.80 - 2.04i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.758 - 2.33i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.24705983540410762669513899140, −10.37176700842121055547008017874, −9.395588177566629892500695043825, −8.385137105860356243938368676786, −7.892716723715126845701137572990, −6.56428116469734175851869716128, −5.37038647393993493425842138095, −4.05809881548241987104843882082, −2.04063584341240218891677689562, −1.29728451353331249801466750643, 1.24229444944271261465591660443, 3.59593659762273429428076372639, 4.69213924341779913792347105502, 6.02504053883970364598613316144, 6.92074521476603065014123750637, 8.063678636074351297307890526710, 8.875337554795964543493206837785, 9.232177112991334886179495659834, 10.65738380037419575243496320456, 11.33002014850668478788161507885

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