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Properties

Label 2-375-25.4-c1-0-12
Degree $2$
Conductor $375$
Sign $0.471 + 0.881i$
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  + (0.951 + 0.309i)2-s + (−0.587 + 0.809i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)6-s − 4.47i·7-s + (−1.76 − 2.42i)8-s + (−0.309 − 0.951i)9-s + (−0.381 + 1.17i)11-s + (0.951 − 0.309i)12-s + (5.34 − 1.73i)13-s + (1.38 − 4.25i)14-s + (−0.309 − 0.951i)16-s + (−2.26 − 3.11i)17-s − i·18-s + (1 − 0.726i)19-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (−0.339 + 0.467i)3-s + (−0.404 − 0.293i)4-s + (−0.330 + 0.239i)6-s − 1.69i·7-s + (−0.623 − 0.858i)8-s + (−0.103 − 0.317i)9-s + (−0.115 + 0.354i)11-s + (0.274 − 0.0892i)12-s + (1.48 − 0.481i)13-s + (0.369 − 1.13i)14-s + (−0.0772 − 0.237i)16-s + (−0.549 − 0.756i)17-s − 0.235i·18-s + (0.229 − 0.166i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.471 + 0.881i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.471 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14319 - 0.685306i\)
\(L(\frac12)\) \(\approx\) \(1.14319 - 0.685306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T + (1.61 + 1.17i)T^{2} \)
7 \( 1 + 4.47iT - 7T^{2} \)
11 \( 1 + (0.381 - 1.17i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-5.34 + 1.73i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.26 + 3.11i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1 + 0.726i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.25 + 1.38i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-5.35 - 3.88i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.23 - 1.62i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.93 + 0.954i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.11 + 3.44i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.70iT - 43T^{2} \)
47 \( 1 + (0.449 - 0.618i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.12 - 2.92i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.23 - 3.80i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.5 + 1.53i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (0.449 + 0.618i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-4.23 - 3.07i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-7.69 - 2.5i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.33 - 10.0i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.66 - 5.11i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-1.26 + 1.73i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.96326090358729352609165985110, −10.45408662348576834088571278667, −9.590109738107885686226315894467, −8.485745644405159967763368586292, −7.14871524304543630468327868809, −6.31475642129971390021626882952, −5.16619186409693410513686881850, −4.24488227509477733320519367732, −3.52571972829921336727536224199, −0.803430060524521198873504055207, 2.07367995342400592765986513129, 3.36478494992875225967949960631, 4.64075556118722235473432436070, 5.90146263466581109795211149832, 6.18889852181922098892269199347, 8.155224478113537325097025515784, 8.542943487364513593360452253441, 9.550907820324029210392855278397, 11.14369042913660955642023592844, 11.67759816611181588700857322143

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