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Properties

Label 2-155-155.3-c1-0-5
Degree $2$
Conductor $155$
Sign $0.850 + 0.525i$
Analytic cond. $1.23768$
Root an. cond. $1.11251$
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  + (0.277 − 1.75i)2-s + (1.23 + 1.53i)3-s + (−1.09 − 0.356i)4-s + (−0.424 + 2.19i)5-s + (3.03 − 1.74i)6-s + (0.165 − 0.255i)7-s + (0.681 − 1.33i)8-s + (−0.183 + 0.861i)9-s + (3.73 + 1.35i)10-s + (−0.000330 + 0.000297i)11-s + (−0.815 − 2.12i)12-s + (0.335 − 0.875i)13-s + (−0.401 − 0.361i)14-s + (−3.88 + 2.07i)15-s + (−4.02 − 2.92i)16-s + (−1.70 + 0.0891i)17-s + ⋯
L(s)  = 1  + (0.196 − 1.24i)2-s + (0.715 + 0.883i)3-s + (−0.549 − 0.178i)4-s + (−0.189 + 0.981i)5-s + (1.23 − 0.714i)6-s + (0.0625 − 0.0963i)7-s + (0.240 − 0.472i)8-s + (−0.0610 + 0.287i)9-s + (1.18 + 0.428i)10-s + (−9.97e−5 + 8.98e−5i)11-s + (−0.235 − 0.613i)12-s + (0.0931 − 0.242i)13-s + (−0.107 − 0.0965i)14-s + (−1.00 + 0.535i)15-s + (−1.00 − 0.731i)16-s + (−0.412 + 0.0216i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ 0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45730 - 0.414160i\)
\(L(\frac12)\) \(\approx\) \(1.45730 - 0.414160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.424 - 2.19i)T \)
31 \( 1 + (-4.07 - 3.79i)T \)
good2 \( 1 + (-0.277 + 1.75i)T + (-1.90 - 0.618i)T^{2} \)
3 \( 1 + (-1.23 - 1.53i)T + (-0.623 + 2.93i)T^{2} \)
7 \( 1 + (-0.165 + 0.255i)T + (-2.84 - 6.39i)T^{2} \)
11 \( 1 + (0.000330 - 0.000297i)T + (1.14 - 10.9i)T^{2} \)
13 \( 1 + (-0.335 + 0.875i)T + (-9.66 - 8.69i)T^{2} \)
17 \( 1 + (1.70 - 0.0891i)T + (16.9 - 1.77i)T^{2} \)
19 \( 1 + (0.902 - 2.02i)T + (-12.7 - 14.1i)T^{2} \)
23 \( 1 + (7.40 + 3.77i)T + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (3.18 - 2.31i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + (-1.59 + 5.96i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.464 - 4.42i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (-6.97 + 2.67i)T + (31.9 - 28.7i)T^{2} \)
47 \( 1 + (8.24 - 1.30i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-4.69 + 3.05i)T + (21.5 - 48.4i)T^{2} \)
59 \( 1 + (6.40 + 0.673i)T + (57.7 + 12.2i)T^{2} \)
61 \( 1 - 7.55iT - 61T^{2} \)
67 \( 1 + (0.735 + 2.74i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.625 + 0.132i)T + (64.8 + 28.8i)T^{2} \)
73 \( 1 + (-15.4 - 0.810i)T + (72.6 + 7.63i)T^{2} \)
79 \( 1 + (-5.91 + 6.57i)T + (-8.25 - 78.5i)T^{2} \)
83 \( 1 + (8.88 + 7.19i)T + (17.2 + 81.1i)T^{2} \)
89 \( 1 + (-2.32 + 7.14i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.79 + 3.52i)T + (-57.0 + 78.4i)T^{2} \)
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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

−12.66770258849941109473643758108, −11.71587364897352319251540015887, −10.63482613278407994327948534633, −10.22364393333424420073780751986, −9.181031418513870945177247094762, −7.82468137456424734412563997018, −6.40582535991678007236328656079, −4.31933453939565183890115056681, −3.50496575516918339695717011193, −2.39992657271917660499496008277, 2.01767177041365805923119908348, 4.34822130281703792204300586980, 5.63322825700479141882565120537, 6.79623880464698201509649226819, 7.900559721369613737545479926218, 8.323975389768183308556050190505, 9.485731978003092722315238885279, 11.29482784960979674917465106339, 12.38387458277611269823762432146, 13.53429381298925357521338141829

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