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Properties

Label 2-2385-265.52-c0-0-4
Degree $2$
Conductor $2385$
Sign $0.379 - 0.925i$
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  + (0.437 − 0.437i)2-s + 0.618i·4-s + (−0.156 + 0.987i)5-s + (1.26 + 1.26i)7-s + (0.707 + 0.707i)8-s + (0.363 + 0.5i)10-s + (0.221 − 0.221i)13-s + 1.10·14-s + (−0.610 − 0.0966i)20-s + (−1.34 − 1.34i)23-s + (−0.951 − 0.309i)25-s − 0.193i·26-s + (−0.778 + 0.778i)28-s + (−0.707 + 0.707i)32-s + (−1.44 + 1.04i)35-s + ⋯
L(s)  = 1  + (0.437 − 0.437i)2-s + 0.618i·4-s + (−0.156 + 0.987i)5-s + (1.26 + 1.26i)7-s + (0.707 + 0.707i)8-s + (0.363 + 0.5i)10-s + (0.221 − 0.221i)13-s + 1.10·14-s + (−0.610 − 0.0966i)20-s + (−1.34 − 1.34i)23-s + (−0.951 − 0.309i)25-s − 0.193i·26-s + (−0.778 + 0.778i)28-s + (−0.707 + 0.707i)32-s + (−1.44 + 1.04i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2385\)    =    \(3^{2} \cdot 5 \cdot 53\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(1.19027\)
Root analytic conductor: \(1.09099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2385} (847, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2385,\ (\ :0),\ 0.379 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.687757719\)
\(L(\frac12)\) \(\approx\) \(1.687757719\)
\(L(1)\) 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 \( 1 + (0.156 - 0.987i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-0.437 + 0.437i)T - iT^{2} \)
7 \( 1 + (-1.26 - 1.26i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-0.221 + 0.221i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (1.34 + 1.34i)T + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.642 + 0.642i)T + iT^{2} \)
41 \( 1 + 1.78iT - T^{2} \)
43 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 0.312iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.831 - 0.831i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.39 - 1.39i)T + iT^{2} \)
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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

−9.055273796735032545594768499325, −8.481529284171430259071170163126, −7.82500924155946355986077508406, −7.14471598201569218059913294655, −6.01551838654121387425615683442, −5.33647627612133949719810826209, −4.32517546098766729924469453015, −3.61169528806742828997469086691, −2.36797251755151925912483369283, −2.17730721520262384172828419723, 1.10274054816285054716898597743, 1.74865919950082576267216599250, 3.76463554041042776346932577052, 4.40242660934973347767687651986, 4.96108457612124592451338789844, 5.74486711442119976075857777817, 6.61654900199044980084742111939, 7.73575353949359439774309134815, 7.84622385284706663550499115894, 8.973052293706699991244518575880

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