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Properties

Label 2-605-11.3-c1-0-12
Degree $2$
Conductor $605$
Sign $-0.0915 - 0.995i$
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  + (1.11 − 0.806i)2-s + (1.00 + 3.10i)3-s + (−0.0357 + 0.110i)4-s + (−0.809 − 0.587i)5-s + (3.62 + 2.63i)6-s + (0.424 − 1.30i)7-s + (0.897 + 2.76i)8-s + (−6.18 + 4.49i)9-s − 1.37·10-s − 0.377·12-s + (−1.56 + 1.13i)13-s + (−0.582 − 1.79i)14-s + (1.00 − 3.10i)15-s + (3.03 + 2.20i)16-s + (5.02 + 3.64i)17-s + (−3.24 + 9.97i)18-s + ⋯
L(s)  = 1  + (0.785 − 0.570i)2-s + (0.581 + 1.79i)3-s + (−0.0178 + 0.0550i)4-s + (−0.361 − 0.262i)5-s + (1.47 + 1.07i)6-s + (0.160 − 0.493i)7-s + (0.317 + 0.976i)8-s + (−2.06 + 1.49i)9-s − 0.434·10-s − 0.108·12-s + (−0.433 + 0.315i)13-s + (−0.155 − 0.478i)14-s + (0.260 − 0.800i)15-s + (0.759 + 0.551i)16-s + (1.21 + 0.885i)17-s + (−0.763 + 2.35i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0915 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0915 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.0915 - 0.995i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.0915 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54992 + 1.69890i\)
\(L(\frac12)\) \(\approx\) \(1.54992 + 1.69890i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-1.11 + 0.806i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-1.00 - 3.10i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.424 + 1.30i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.56 - 1.13i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.02 - 3.64i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.251 + 0.773i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.63T + 23T^{2} \)
29 \( 1 + (-2.42 + 7.45i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.75 + 2.00i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.39 - 4.30i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.564 + 1.73i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.46T + 43T^{2} \)
47 \( 1 + (-1.46 - 4.50i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-6.79 + 4.93i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.344 + 1.06i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.05 + 1.49i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.73T + 67T^{2} \)
71 \( 1 + (-0.902 - 0.655i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.83 + 11.8i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.69 + 3.41i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-12.9 - 9.39i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 2.70T + 89T^{2} \)
97 \( 1 + (11.4 - 8.32i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.79801806142950797722377226168, −10.16824340085426335852743185929, −9.352522589031199546349348434237, −8.259382512357658561732128199036, −7.83181234014040987380619637567, −5.80660647770510826428247715038, −4.80864239297183981159254558670, −4.14696175258750770863867040937, −3.57336343725127650670637744912, −2.45138474460559544286687076862, 0.992393588615408681186815768482, 2.49964447196955977072696604794, 3.57035361862323215281865744425, 5.22159075846315396004856642851, 5.96111171720857358898041082180, 6.95566417765314591052873002346, 7.44781870237928854193771721321, 8.289223212554556313534445919515, 9.291148549070024166221840489756, 10.46633994631079420663786780818

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