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Properties

Label 2-2385-265.179-c0-0-0
Degree $2$
Conductor $2385$
Sign $-0.915 + 0.402i$
Analytic cond. $1.19027$
Root an. cond. $1.09099$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.239 + 1.30i)2-s + (−0.709 + 0.269i)4-s + (−0.787 + 0.616i)5-s + (0.165 + 0.273i)8-s + (−0.992 − 0.879i)10-s + (−0.885 + 0.784i)16-s + (−1.93 + 0.477i)17-s + (−0.542 + 0.244i)19-s + (0.392 − 0.649i)20-s + (1.25 + 1.25i)23-s + (0.239 − 0.970i)25-s + (−0.344 − 0.107i)31-s + (−0.983 − 0.770i)32-s + (−1.08 − 2.41i)34-s + (−0.448 − 0.649i)38-s + ⋯
L(s)  = 1  + (0.239 + 1.30i)2-s + (−0.709 + 0.269i)4-s + (−0.787 + 0.616i)5-s + (0.165 + 0.273i)8-s + (−0.992 − 0.879i)10-s + (−0.885 + 0.784i)16-s + (−1.93 + 0.477i)17-s + (−0.542 + 0.244i)19-s + (0.392 − 0.649i)20-s + (1.25 + 1.25i)23-s + (0.239 − 0.970i)25-s + (−0.344 − 0.107i)31-s + (−0.983 − 0.770i)32-s + (−1.08 − 2.41i)34-s + (−0.448 − 0.649i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2385\)    =    \(3^{2} \cdot 5 \cdot 53\)
Sign: $-0.915 + 0.402i$
Analytic conductor: \(1.19027\)
Root analytic conductor: \(1.09099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2385} (1504, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2385,\ (\ :0),\ -0.915 + 0.402i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8830331751\)
\(L(\frac12)\) \(\approx\) \(0.8830331751\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.787 - 0.616i)T \)
53 \( 1 + (0.517 - 0.855i)T \)
good2 \( 1 + (-0.239 - 1.30i)T + (-0.935 + 0.354i)T^{2} \)
7 \( 1 + (-0.354 - 0.935i)T^{2} \)
11 \( 1 + (-0.568 - 0.822i)T^{2} \)
13 \( 1 + (0.748 + 0.663i)T^{2} \)
17 \( 1 + (1.93 - 0.477i)T + (0.885 - 0.464i)T^{2} \)
19 \( 1 + (0.542 - 0.244i)T + (0.663 - 0.748i)T^{2} \)
23 \( 1 + (-1.25 - 1.25i)T + iT^{2} \)
29 \( 1 + (-0.568 + 0.822i)T^{2} \)
31 \( 1 + (0.344 + 0.107i)T + (0.822 + 0.568i)T^{2} \)
37 \( 1 + (0.120 - 0.992i)T^{2} \)
41 \( 1 + (-0.822 + 0.568i)T^{2} \)
43 \( 1 + (0.120 + 0.992i)T^{2} \)
47 \( 1 + (0.814 - 0.0989i)T + (0.970 - 0.239i)T^{2} \)
59 \( 1 + (0.970 - 0.239i)T^{2} \)
61 \( 1 + (0.885 - 0.535i)T + (0.464 - 0.885i)T^{2} \)
67 \( 1 + (-0.663 - 0.748i)T^{2} \)
71 \( 1 + (0.992 - 0.120i)T^{2} \)
73 \( 1 + (-0.464 - 0.885i)T^{2} \)
79 \( 1 + (-0.283 + 1.54i)T + (-0.935 - 0.354i)T^{2} \)
83 \( 1 + (-1.32 + 1.32i)T - iT^{2} \)
89 \( 1 + (0.885 - 0.464i)T^{2} \)
97 \( 1 + (0.970 + 0.239i)T^{2} \)
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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

−9.132820056004798228695535349086, −8.678249012703092455269598689256, −7.72141401263324774080326120017, −7.30122054767848010849919870894, −6.50405632596202779261194004023, −6.01787847001079674687101653844, −4.84371600442663099110458877669, −4.28712406828641208492695449902, −3.22433025486469460291175337050, −1.99478031841275561431218751250, 0.52601406899038369302046951507, 1.92117541109124773072043756707, 2.81105361222663386250816415976, 3.78068482060397298208222838929, 4.57504960558875934854441287467, 5.00452051983255243214296090172, 6.60768046543073857662998970048, 7.05760322886142868587344496820, 8.236266740940774802530259467600, 8.895054642546187822685013353645

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