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Properties

Label 2-952-56.37-c1-0-25
Degree $2$
Conductor $952$
Sign $0.126 - 0.991i$
Analytic cond. $7.60175$
Root an. cond. $2.75712$
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  + (−1.22 + 0.707i)2-s + (0.275 − 0.158i)3-s + (0.999 − 1.73i)4-s + (−1.22 − 0.707i)5-s + (−0.224 + 0.389i)6-s + (−2 − 1.73i)7-s + 2.82i·8-s + (−1.44 + 2.51i)9-s + 2·10-s + (−0.275 + 0.158i)11-s − 0.635i·12-s − 2.51i·13-s + (3.67 + 0.707i)14-s − 0.449·15-s + (−2.00 − 3.46i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (0.158 − 0.0917i)3-s + (0.499 − 0.866i)4-s + (−0.547 − 0.316i)5-s + (−0.0917 + 0.158i)6-s + (−0.755 − 0.654i)7-s + 0.999i·8-s + (−0.483 + 0.836i)9-s + 0.632·10-s + (−0.0829 + 0.0479i)11-s − 0.183i·12-s − 0.696i·13-s + (0.981 + 0.188i)14-s − 0.116·15-s + (−0.500 − 0.866i)16-s + (0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(952\)    =    \(2^{3} \cdot 7 \cdot 17\)
Sign: $0.126 - 0.991i$
Analytic conductor: \(7.60175\)
Root analytic conductor: \(2.75712\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{952} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 952,\ (\ :1/2),\ 0.126 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.481208 + 0.423777i\)
\(L(\frac12)\) \(\approx\) \(0.481208 + 0.423777i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
7 \( 1 + (2 + 1.73i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.275 + 0.158i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.22 + 0.707i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.275 - 0.158i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.51iT - 13T^{2} \)
19 \( 1 + (-6 - 3.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.87iT - 29T^{2} \)
31 \( 1 + (3.44 + 5.97i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.34 - 4.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.34T + 41T^{2} \)
43 \( 1 - 7.70iT - 43T^{2} \)
47 \( 1 + (2.44 - 4.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.72 + 3.30i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.22 + 4.17i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.34 - 2.51i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 + (-6.22 - 10.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.05 + 1.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 + (6.39 - 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−9.862884877022682663368555844189, −9.604221013961453802710309473094, −8.204845062948259139135865220298, −7.889584610325606247248317080508, −7.18650948702057845692245947837, −5.96865434171928277056726804475, −5.31174550649267067614243408877, −3.91634924473445022482679498971, −2.69842406090971806462261920673, −1.08845522525120845197712319335, 0.46186646110868511154285772002, 2.38526100574287342550872140980, 3.22478047913895809189945311442, 4.05871142094889206934151636753, 5.70964866069682878920233431846, 6.70016033639347155425016912764, 7.38331456286771723061072970428, 8.419942226424659331855119929624, 9.222184250974530347173618557177, 9.566606933554789038331932404395

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