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Properties

Label 2-2475-165.98-c0-0-3
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  + (−0.541 + 0.541i)2-s + 0.414i·4-s + (−0.541 − 0.541i)7-s + (−0.765 − 0.765i)8-s − i·11-s + (−1.30 + 1.30i)13-s + 0.585·14-s + 0.414·16-s + (1.30 − 1.30i)17-s + (0.541 + 0.541i)22-s − 1.41i·26-s + (0.224 − 0.224i)28-s + 1.41·31-s + (0.541 − 0.541i)32-s + 1.41i·34-s + ⋯
L(s)  = 1  + (−0.541 + 0.541i)2-s + 0.414i·4-s + (−0.541 − 0.541i)7-s + (−0.765 − 0.765i)8-s − i·11-s + (−1.30 + 1.30i)13-s + 0.585·14-s + 0.414·16-s + (1.30 − 1.30i)17-s + (0.541 + 0.541i)22-s − 1.41i·26-s + (0.224 − 0.224i)28-s + 1.41·31-s + (0.541 − 0.541i)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

\(L(\frac{1}{2})\) \(\approx\) \(0.7068073600\)
\(L(\frac12)\) \(\approx\) \(0.7068073600\)
\(L(1)\) 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 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
7 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
13 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
17 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
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 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
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 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
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

−9.101966751580530262624418136206, −8.298890539732595094360926398626, −7.44018526927773421122582562065, −7.05307290886927627005272307803, −6.29480912950621057386276866568, −5.29262338955193534011268419229, −4.25218139071012591622617209653, −3.36243776231846093117866005031, −2.55063852125601423670944732840, −0.64989712838208429289733255131, 1.15422160177914600974181437256, 2.41619699996674278332554062167, 2.99042855642324379115767583228, 4.35180455501678096192345166029, 5.40735582901942160152747633534, 5.83205943049423448607340609782, 6.84121350017411392575639705758, 7.85630468098462631103507763792, 8.361592554200768329866557670569, 9.468780575126623163262100552763

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