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Properties

Label 2-2475-165.98-c0-0-7
Degree $2$
Conductor $2475$
Sign $-0.998 - 0.0618i$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
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  + (1.30 − 1.30i)2-s − 2.41i·4-s + (−1.30 − 1.30i)7-s + (−1.84 − 1.84i)8-s + i·11-s + (0.541 − 0.541i)13-s − 3.41·14-s − 2.41·16-s + (0.541 − 0.541i)17-s + (1.30 + 1.30i)22-s − 1.41i·26-s + (−3.15 + 3.15i)28-s − 1.41·31-s + (−1.30 + 1.30i)32-s − 1.41i·34-s + ⋯
L(s)  = 1  + (1.30 − 1.30i)2-s − 2.41i·4-s + (−1.30 − 1.30i)7-s + (−1.84 − 1.84i)8-s + i·11-s + (0.541 − 0.541i)13-s − 3.41·14-s − 2.41·16-s + (0.541 − 0.541i)17-s + (1.30 + 1.30i)22-s − 1.41i·26-s + (−3.15 + 3.15i)28-s − 1.41·31-s + (−1.30 + 1.30i)32-s − 1.41i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.998 - 0.0618i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ -0.998 - 0.0618i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.934253023\)
\(L(\frac12)\) \(\approx\) \(1.934253023\)
\(L(1)\) 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 \( 1 \)
11 \( 1 - iT \)
good2 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
7 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
13 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
17 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.327108540409619408875573161914, −7.74725052697953131109155387923, −6.92204320920041992167446866606, −6.23088965705443763814858456302, −5.30354975225705935400873497960, −4.50452367822592847348526047994, −3.60693317001582164000767966538, −3.28777253491558199089231990994, −2.07865682220562342257641005650, −0.836081933601431849760190620280, 2.39598001859442092716101897241, 3.47686432292660095123866410917, 3.73879156316811042931091016126, 5.17464033832320990584218340242, 5.69672194746190355736743595421, 6.28082134706015807178211592237, 6.80376972586997397423991418281, 7.80250098486903735247828208918, 8.700699516796944811395102030361, 8.984137398700288985435572185700

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