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Properties

Label 2-605-55.32-c1-0-39
Degree $2$
Conductor $605$
Sign $-0.00560 + 0.999i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.738 − 0.738i)2-s + (1.99 − 1.99i)3-s + 0.908i·4-s + (0.742 − 2.10i)5-s − 2.94i·6-s + (0.388 − 0.388i)7-s + (2.14 + 2.14i)8-s − 4.93i·9-s + (−1.01 − 2.10i)10-s + (1.80 + 1.80i)12-s + (2.29 + 2.29i)13-s − 0.574i·14-s + (−2.72 − 5.67i)15-s + 1.35·16-s + (−4.00 + 4.00i)17-s + (−3.64 − 3.64i)18-s + ⋯
L(s)  = 1  + (0.522 − 0.522i)2-s + (1.14 − 1.14i)3-s + 0.454i·4-s + (0.331 − 0.943i)5-s − 1.20i·6-s + (0.146 − 0.146i)7-s + (0.759 + 0.759i)8-s − 1.64i·9-s + (−0.319 − 0.666i)10-s + (0.522 + 0.522i)12-s + (0.636 + 0.636i)13-s − 0.153i·14-s + (−0.702 − 1.46i)15-s + 0.339·16-s + (−0.971 + 0.971i)17-s + (−0.858 − 0.858i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00560 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00560 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.00560 + 0.999i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.00560 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09347 - 2.10524i\)
\(L(\frac12)\) \(\approx\) \(2.09347 - 2.10524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.742 + 2.10i)T \)
11 \( 1 \)
good2 \( 1 + (-0.738 + 0.738i)T - 2iT^{2} \)
3 \( 1 + (-1.99 + 1.99i)T - 3iT^{2} \)
7 \( 1 + (-0.388 + 0.388i)T - 7iT^{2} \)
13 \( 1 + (-2.29 - 2.29i)T + 13iT^{2} \)
17 \( 1 + (4.00 - 4.00i)T - 17iT^{2} \)
19 \( 1 + 1.55T + 19T^{2} \)
23 \( 1 + (0.803 - 0.803i)T - 23iT^{2} \)
29 \( 1 + 4.25T + 29T^{2} \)
31 \( 1 + 1.64T + 31T^{2} \)
37 \( 1 + (-0.676 - 0.676i)T + 37iT^{2} \)
41 \( 1 + 8.94iT - 41T^{2} \)
43 \( 1 + (-2.55 - 2.55i)T + 43iT^{2} \)
47 \( 1 + (2.87 + 2.87i)T + 47iT^{2} \)
53 \( 1 + (-4.98 + 4.98i)T - 53iT^{2} \)
59 \( 1 - 6.76iT - 59T^{2} \)
61 \( 1 - 9.20iT - 61T^{2} \)
67 \( 1 + (2.62 + 2.62i)T + 67iT^{2} \)
71 \( 1 - 6.85T + 71T^{2} \)
73 \( 1 + (7.13 + 7.13i)T + 73iT^{2} \)
79 \( 1 - 4.16T + 79T^{2} \)
83 \( 1 + (8.01 + 8.01i)T + 83iT^{2} \)
89 \( 1 + 3.64iT - 89T^{2} \)
97 \( 1 + (-11.7 - 11.7i)T + 97iT^{2} \)
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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

−10.61177869428053039186335332934, −9.093578958789497198052276435409, −8.679395151598662953358455405307, −7.936281904705712520672581426614, −7.06068429132085515017199224511, −5.91250003723255436326960122635, −4.43113526862787990451610698281, −3.65353699734518317378606599109, −2.25513926817652699438777055895, −1.59847070455709017931526172411, 2.22339209825525684740975842408, 3.33575767837908553610391857731, 4.28624967952015835331121846546, 5.26165526816285981655607598205, 6.27038213856730902353439926244, 7.24696281377083500992928552040, 8.295257473687485002230851655995, 9.319853353982397380237530552847, 9.891281179981733143202054794603, 10.72044956140530099021358108218

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