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Properties

Label 2-154-7.4-c1-0-0
Degree $2$
Conductor $154$
Sign $-0.832 - 0.553i$
Analytic cond. $1.22969$
Root an. cond. $1.10891$
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  + (0.5 + 0.866i)2-s + (−1.32 + 2.29i)3-s + (−0.499 + 0.866i)4-s + (0.822 + 1.42i)5-s − 2.64·6-s + (−1.32 − 2.29i)7-s − 0.999·8-s + (−2 − 3.46i)9-s + (−0.822 + 1.42i)10-s + (−0.5 + 0.866i)11-s + (−1.32 − 2.29i)12-s + 5·13-s + (1.32 − 2.29i)14-s − 4.35·15-s + (−0.5 − 0.866i)16-s + (−3 + 5.19i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.763 + 1.32i)3-s + (−0.249 + 0.433i)4-s + (0.368 + 0.637i)5-s − 1.08·6-s + (−0.499 − 0.866i)7-s − 0.353·8-s + (−0.666 − 1.15i)9-s + (−0.260 + 0.450i)10-s + (−0.150 + 0.261i)11-s + (−0.381 − 0.661i)12-s + 1.38·13-s + (0.353 − 0.612i)14-s − 1.12·15-s + (−0.125 − 0.216i)16-s + (−0.727 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $-0.832 - 0.553i$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ -0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.295066 + 0.976348i\)
\(L(\frac12)\) \(\approx\) \(0.295066 + 0.976348i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.32 + 2.29i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (1.32 - 2.29i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.822 - 1.42i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.82 - 4.88i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.822 + 1.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.29T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.82 + 3.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (1.35 + 2.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.822 - 1.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.32 - 4.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.14 + 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.96 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.35T + 71T^{2} \)
73 \( 1 + (0.177 - 0.306i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.32 - 2.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.70T + 83T^{2} \)
89 \( 1 + (3.29 + 5.70i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−13.60699235512231423597058784253, −12.43743299112825979912552603317, −10.96173829168305793906722044289, −10.49096757214485104510289893447, −9.565615175585211110026787720482, −8.134117974988046124052436441104, −6.50950775462488720817289925770, −5.93403474009816346402816364224, −4.41205116391757646211617602553, −3.57450956170679545052740211294, 1.08928941804182991840182327617, 2.77152966487750449609004297313, 5.02323960448957484868037720839, 5.96478698770643686587908090181, 6.88983008761910233471025359542, 8.550679611979928545023305608530, 9.430245649503273250634322470650, 11.06645592605434790653283615713, 11.72042628501259952489798890959, 12.57413659309207540298850665839

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