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Properties

Label 2-152-152.107-c1-0-2
Degree $2$
Conductor $152$
Sign $0.00729 - 0.999i$
Analytic cond. $1.21372$
Root an. cond. $1.10169$
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  + (−1.34 − 0.445i)2-s + (0.705 − 0.407i)3-s + (1.60 + 1.19i)4-s + (−3.59 + 2.07i)5-s + (−1.12 + 0.232i)6-s + 3.24i·7-s + (−1.62 − 2.31i)8-s + (−1.16 + 2.02i)9-s + (5.75 − 1.18i)10-s − 1.21·11-s + (1.61 + 0.190i)12-s + (2.01 − 3.48i)13-s + (1.44 − 4.35i)14-s + (−1.69 + 2.93i)15-s + (1.14 + 3.83i)16-s + (1.50 + 2.60i)17-s + ⋯
L(s)  = 1  + (−0.949 − 0.314i)2-s + (0.407 − 0.235i)3-s + (0.801 + 0.597i)4-s + (−1.60 + 0.928i)5-s + (−0.460 + 0.0949i)6-s + 1.22i·7-s + (−0.572 − 0.819i)8-s + (−0.389 + 0.674i)9-s + (1.81 − 0.375i)10-s − 0.367·11-s + (0.467 + 0.0548i)12-s + (0.558 − 0.967i)13-s + (0.386 − 1.16i)14-s + (−0.436 + 0.756i)15-s + (0.285 + 0.958i)16-s + (0.364 + 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00729 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00729 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.00729 - 0.999i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.00729 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.371224 + 0.368524i\)
\(L(\frac12)\) \(\approx\) \(0.371224 + 0.368524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.445i)T \)
19 \( 1 + (2.16 - 3.78i)T \)
good3 \( 1 + (-0.705 + 0.407i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (3.59 - 2.07i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.24iT - 7T^{2} \)
11 \( 1 + 1.21T + 11T^{2} \)
13 \( 1 + (-2.01 + 3.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.50 - 2.60i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.01 - 1.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.35 + 4.08i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.99T + 31T^{2} \)
37 \( 1 + 1.34T + 37T^{2} \)
41 \( 1 + (-2.97 + 1.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.01 + 1.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.37 - 4.83i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.22 - 2.11i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.35 - 5.39i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.300 - 0.173i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.31 - 4.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.581 - 1.00i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.46 - 2.54i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.57 - 4.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (12.1 + 7.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.1 + 6.99i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.80894661075967560821510419051, −12.05472700233723356633646897329, −11.06372571457182749611428324216, −10.48496575960992880225762343203, −8.780396353696957132797661488650, −8.073833496682918007920508973387, −7.47490476725979047566029533509, −5.91572844611268353411751430834, −3.55590699492925666858786280281, −2.55129547066609802181423236233, 0.66099482410548465206698842000, 3.52741753331325522858828259881, 4.76628601478770529148610355129, 6.80420313523100786089575831203, 7.63988543341374083061547221415, 8.666603240281620556849279832500, 9.257254205314238584090423845499, 10.78674277499961853759049709332, 11.46564134048262891580913802131, 12.52087682277748283403000352245

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