Properties

Label 2-605-55.52-c1-0-1
Degree $2$
Conductor $605$
Sign $0.954 + 0.298i$
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  + (0.396 − 2.50i)2-s + (−1.78 − 0.910i)3-s + (−4.21 − 1.36i)4-s + (−2.22 + 0.188i)5-s + (−2.98 + 4.11i)6-s + (0.820 + 1.61i)7-s + (−2.79 + 5.48i)8-s + (0.599 + 0.824i)9-s + (−0.412 + 5.65i)10-s + (6.27 + 6.27i)12-s + (−2.22 − 0.352i)13-s + (4.35 − 1.41i)14-s + (4.15 + 1.69i)15-s + (5.46 + 3.96i)16-s + (3.32 − 0.526i)17-s + (2.30 − 1.17i)18-s + ⋯
L(s)  = 1  + (0.280 − 1.77i)2-s + (−1.03 − 0.525i)3-s + (−2.10 − 0.684i)4-s + (−0.996 + 0.0840i)5-s + (−1.21 + 1.67i)6-s + (0.310 + 0.608i)7-s + (−0.987 + 1.93i)8-s + (0.199 + 0.274i)9-s + (−0.130 + 1.78i)10-s + (1.81 + 1.81i)12-s + (−0.617 − 0.0978i)13-s + (1.16 − 0.378i)14-s + (1.07 + 0.436i)15-s + (1.36 + 0.991i)16-s + (0.805 − 0.127i)17-s + (0.542 − 0.276i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.954 + 0.298i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (602, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.954 + 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252952 - 0.0386081i\)
\(L(\frac12)\) \(\approx\) \(0.252952 - 0.0386081i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.22 - 0.188i)T \)
11 \( 1 \)
good2 \( 1 + (-0.396 + 2.50i)T + (-1.90 - 0.618i)T^{2} \)
3 \( 1 + (1.78 + 0.910i)T + (1.76 + 2.42i)T^{2} \)
7 \( 1 + (-0.820 - 1.61i)T + (-4.11 + 5.66i)T^{2} \)
13 \( 1 + (2.22 + 0.352i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (-3.32 + 0.526i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (0.403 + 1.24i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.16 - 3.16i)T - 23iT^{2} \)
29 \( 1 + (0.907 - 2.79i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.24 + 2.35i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.85 - 1.45i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (1.23 - 0.402i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (4.55 + 4.55i)T + 43iT^{2} \)
47 \( 1 + (3.50 - 6.88i)T + (-27.6 - 38.0i)T^{2} \)
53 \( 1 + (0.996 - 6.29i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-11.7 - 3.82i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.47 - 7.53i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.46 + 2.46i)T + 67iT^{2} \)
71 \( 1 + (-5.47 - 3.98i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.41 - 4.28i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-0.926 + 0.673i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.63 - 10.3i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 - 3.85iT - 89T^{2} \)
97 \( 1 + (-0.768 - 0.121i)T + (92.2 + 29.9i)T^{2} \)
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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

−10.95127717584198950573140307210, −10.16745825438931438555779001158, −9.155382911611849993077596562781, −8.140281646387641383600663686656, −7.02372765905960978801642407167, −5.62663859448035394887258539600, −4.90420116486672450376934732002, −3.77180441868517194422215943704, −2.68547447462953514083907365233, −1.21792764829839554741926918618, 0.17185663139096250433403578480, 3.72954512570845808227260210672, 4.60003612955972789521145157845, 5.15514711176907737098581288534, 6.18460435704190212450476948824, 7.03529762021410508023348449811, 7.897338344821671745836109132569, 8.424739326509791396085189914414, 9.794956845778261069640435630364, 10.60658208407431752757795474542

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