Properties

Label 2-145-145.108-c1-0-3
Degree $2$
Conductor $145$
Sign $-0.626 - 0.779i$
Analytic cond. $1.15783$
Root an. cond. $1.07602$
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  + (1.18 + 2.46i)2-s + (0.229 + 0.287i)3-s + (−3.41 + 4.27i)4-s + (1.74 − 1.39i)5-s + (−0.436 + 0.906i)6-s + (0.598 + 0.0674i)7-s + (−9.24 − 2.11i)8-s + (0.637 − 2.79i)9-s + (5.50 + 2.65i)10-s + (−0.659 − 0.414i)11-s − 2.01·12-s + (−3.60 − 2.26i)13-s + (0.543 + 1.55i)14-s + (0.802 + 0.183i)15-s + (−3.33 − 14.6i)16-s + 5.10i·17-s + ⋯
L(s)  = 1  + (0.838 + 1.74i)2-s + (0.132 + 0.166i)3-s + (−1.70 + 2.13i)4-s + (0.782 − 0.623i)5-s + (−0.178 + 0.370i)6-s + (0.226 + 0.0254i)7-s + (−3.26 − 0.746i)8-s + (0.212 − 0.930i)9-s + (1.74 + 0.839i)10-s + (−0.198 − 0.124i)11-s − 0.581·12-s + (−1.00 − 0.628i)13-s + (0.145 + 0.415i)14-s + (0.207 + 0.0474i)15-s + (−0.833 − 3.65i)16-s + 1.23i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.626 - 0.779i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :1/2),\ -0.626 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.711017 + 1.48489i\)
\(L(\frac12)\) \(\approx\) \(0.711017 + 1.48489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.74 + 1.39i)T \)
29 \( 1 + (3.59 + 4.00i)T \)
good2 \( 1 + (-1.18 - 2.46i)T + (-1.24 + 1.56i)T^{2} \)
3 \( 1 + (-0.229 - 0.287i)T + (-0.667 + 2.92i)T^{2} \)
7 \( 1 + (-0.598 - 0.0674i)T + (6.82 + 1.55i)T^{2} \)
11 \( 1 + (0.659 + 0.414i)T + (4.77 + 9.91i)T^{2} \)
13 \( 1 + (3.60 + 2.26i)T + (5.64 + 11.7i)T^{2} \)
17 \( 1 - 5.10iT - 17T^{2} \)
19 \( 1 + (-4.17 + 0.470i)T + (18.5 - 4.22i)T^{2} \)
23 \( 1 + (-1.07 - 3.06i)T + (-17.9 + 14.3i)T^{2} \)
31 \( 1 + (2.80 - 8.00i)T + (-24.2 - 19.3i)T^{2} \)
37 \( 1 + (0.835 - 3.66i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (3.74 + 3.74i)T + 41iT^{2} \)
43 \( 1 + (-1.85 - 0.891i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.07 + 4.71i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-7.55 - 2.64i)T + (41.4 + 33.0i)T^{2} \)
59 \( 1 + 10.0iT - 59T^{2} \)
61 \( 1 + (4.16 + 0.469i)T + (59.4 + 13.5i)T^{2} \)
67 \( 1 + (4.92 - 3.09i)T + (29.0 - 60.3i)T^{2} \)
71 \( 1 + (-12.7 + 2.90i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (1.91 - 3.98i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + (3.37 - 2.11i)T + (34.2 - 71.1i)T^{2} \)
83 \( 1 + (2.18 - 0.246i)T + (80.9 - 18.4i)T^{2} \)
89 \( 1 + (-6.44 - 2.25i)T + (69.5 + 55.4i)T^{2} \)
97 \( 1 + (1.60 - 2.01i)T + (-21.5 - 94.5i)T^{2} \)
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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

−13.64153027425287647136138790083, −12.80451422451951963931531908218, −12.11439461492819856579762297441, −9.909256925231776156248107656924, −8.975525118923295449231729827135, −7.999421126438195087277519520890, −6.84507368725467403975919842206, −5.70813001398450695825459815321, −4.95013535458732833741851762976, −3.52104115365515544560149803337, 1.95060965199937645520613627120, 2.89409321848025737771457397853, 4.64783146600406242621332101455, 5.52649590737836023185098663341, 7.27151467036574979984146559742, 9.276089348663961776448412626254, 9.934242762728265604752808368435, 10.90575150518594484281178371124, 11.64655231790409119815226844641, 12.76306114060322285435468657135

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