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Properties

Label 2-2535-195.122-c0-0-0
Degree $2$
Conductor $2535$
Sign $0.606 + 0.794i$
Analytic cond. $1.26512$
Root an. cond. $1.12477$
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  − 1.84i·2-s + (−0.707 + 0.707i)3-s − 2.41·4-s + (0.923 + 0.382i)5-s + (1.30 + 1.30i)6-s + 2.61i·8-s − 1.00i·9-s + (0.707 − 1.70i)10-s + (−0.541 + 0.541i)11-s + (1.70 − 1.70i)12-s + (−0.923 + 0.382i)15-s + 2.41·16-s − 1.84·18-s + (−2.23 − 0.923i)20-s + (1 + i)22-s + ⋯
L(s)  = 1  − 1.84i·2-s + (−0.707 + 0.707i)3-s − 2.41·4-s + (0.923 + 0.382i)5-s + (1.30 + 1.30i)6-s + 2.61i·8-s − 1.00i·9-s + (0.707 − 1.70i)10-s + (−0.541 + 0.541i)11-s + (1.70 − 1.70i)12-s + (−0.923 + 0.382i)15-s + 2.41·16-s − 1.84·18-s + (−2.23 − 0.923i)20-s + (1 + i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $0.606 + 0.794i$
Analytic conductor: \(1.26512\)
Root analytic conductor: \(1.12477\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (2267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :0),\ 0.606 + 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8723975347\)
\(L(\frac12)\) \(\approx\) \(0.8723975347\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 \)
good2 \( 1 + 1.84iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + 0.765T + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + 0.765T + T^{2} \)
89 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
97 \( 1 + T^{2} \)
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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

−9.346995089489309551136413113710, −8.851196307075350136709472598943, −7.58059457160372931421936616964, −6.35287954553924623013584274338, −5.53480379009137217638485905148, −4.79364068818434376873471315870, −4.08975576440504573159946268715, −3.05359915406409167059627026290, −2.34144331836528153262123765733, −1.14543398237975088650199646734, 0.77098809745456040963397846711, 2.34627440202336135034204380815, 4.07095935789127279551150707420, 5.07996021366720875292391976513, 5.60225996314010321812035266638, 6.02051353592083715852148145307, 6.90445238588539392553660687909, 7.39887388570128038639052335358, 8.343713637445366944978878601547, 8.753799537776959147637556875038

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