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Properties

Label 2-155-155.83-c1-0-1
Degree $2$
Conductor $155$
Sign $-0.0906 - 0.995i$
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  + (−1.38 + 0.219i)2-s + (−2.64 − 2.14i)3-s + (−0.0349 + 0.0113i)4-s + (−0.236 − 2.22i)5-s + (4.13 + 2.38i)6-s + (−2.03 + 1.31i)7-s + (2.54 − 1.29i)8-s + (1.79 + 8.42i)9-s + (0.814 + 3.02i)10-s + (−0.152 − 0.137i)11-s + (0.116 + 0.0448i)12-s + (−0.0934 + 0.0358i)13-s + (2.52 − 2.27i)14-s + (−4.14 + 6.39i)15-s + (−3.17 + 2.30i)16-s + (−0.213 + 4.07i)17-s + ⋯
L(s)  = 1  + (−0.978 + 0.154i)2-s + (−1.52 − 1.23i)3-s + (−0.0174 + 0.00567i)4-s + (−0.105 − 0.994i)5-s + (1.68 + 0.974i)6-s + (−0.767 + 0.498i)7-s + (0.898 − 0.458i)8-s + (0.597 + 2.80i)9-s + (0.257 + 0.956i)10-s + (−0.0459 − 0.0413i)11-s + (0.0337 + 0.0129i)12-s + (−0.0259 + 0.00994i)13-s + (0.673 − 0.606i)14-s + (−1.06 + 1.65i)15-s + (−0.793 + 0.576i)16-s + (−0.0517 + 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0906 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0906 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155\)    =    \(5 \cdot 31\)
Sign: $-0.0906 - 0.995i$
Analytic conductor: \(1.23768\)
Root analytic conductor: \(1.11251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{155} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 155,\ (\ :1/2),\ -0.0906 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0502536 + 0.0550347i\)
\(L(\frac12)\) \(\approx\) \(0.0502536 + 0.0550347i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.236 + 2.22i)T \)
31 \( 1 + (5.54 + 0.515i)T \)
good2 \( 1 + (1.38 - 0.219i)T + (1.90 - 0.618i)T^{2} \)
3 \( 1 + (2.64 + 2.14i)T + (0.623 + 2.93i)T^{2} \)
7 \( 1 + (2.03 - 1.31i)T + (2.84 - 6.39i)T^{2} \)
11 \( 1 + (0.152 + 0.137i)T + (1.14 + 10.9i)T^{2} \)
13 \( 1 + (0.0934 - 0.0358i)T + (9.66 - 8.69i)T^{2} \)
17 \( 1 + (0.213 - 4.07i)T + (-16.9 - 1.77i)T^{2} \)
19 \( 1 + (1.08 + 2.43i)T + (-12.7 + 14.1i)T^{2} \)
23 \( 1 + (-0.633 - 1.24i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (-4.86 - 3.53i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (5.01 - 1.34i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.505 + 4.80i)T + (-40.1 - 8.52i)T^{2} \)
43 \( 1 + (4.06 - 10.5i)T + (-31.9 - 28.7i)T^{2} \)
47 \( 1 + (0.377 - 2.38i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-1.45 + 2.24i)T + (-21.5 - 48.4i)T^{2} \)
59 \( 1 + (11.1 - 1.17i)T + (57.7 - 12.2i)T^{2} \)
61 \( 1 - 3.82iT - 61T^{2} \)
67 \( 1 + (5.76 + 1.54i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.12 - 0.876i)T + (64.8 - 28.8i)T^{2} \)
73 \( 1 + (-0.468 - 8.93i)T + (-72.6 + 7.63i)T^{2} \)
79 \( 1 + (-4.80 - 5.33i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (1.33 + 1.64i)T + (-17.2 + 81.1i)T^{2} \)
89 \( 1 + (0.936 + 2.88i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (2.28 + 1.16i)T + (57.0 + 78.4i)T^{2} \)
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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

−12.78897734176267274427257429628, −12.45549923932580115911663201292, −11.23633084817824421557765858942, −10.22092655138138409684591191277, −8.949238643608565586548130464571, −7.994606167779311116634206009380, −6.92837225137325531287497999556, −5.88272546709101337737028194959, −4.71799189870131512352186224974, −1.44920083804026064326952347462, 0.12759511177619499233610700669, 3.63692514333013985981091851478, 4.91013996249962410489022518927, 6.25400042191063653709698109274, 7.24931658234939322271841941128, 9.093365208297285797993078542672, 10.04613624587006610647979395279, 10.39625467411906685068228941301, 11.22003053725832506386766112298, 12.17025966892571333271316199918

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