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Properties

Label 2-950-19.6-c1-0-26
Degree $2$
Conductor $950$
Sign $0.894 + 0.447i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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  + (0.766 − 0.642i)2-s + (3.03 + 1.10i)3-s + (0.173 − 0.984i)4-s + (3.03 − 1.10i)6-s + (1.36 − 2.35i)7-s + (−0.500 − 0.866i)8-s + (5.67 + 4.75i)9-s + (−1.18 − 2.05i)11-s + (1.61 − 2.79i)12-s + (0.0945 − 0.0344i)13-s + (−0.472 − 2.68i)14-s + (−0.939 − 0.342i)16-s + (−4.17 + 3.50i)17-s + 7.40·18-s + (1.18 − 4.19i)19-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (1.74 + 0.636i)3-s + (0.0868 − 0.492i)4-s + (1.23 − 0.450i)6-s + (0.514 − 0.891i)7-s + (−0.176 − 0.306i)8-s + (1.89 + 1.58i)9-s + (−0.358 − 0.621i)11-s + (0.465 − 0.806i)12-s + (0.0262 − 0.00954i)13-s + (−0.126 − 0.716i)14-s + (−0.234 − 0.0855i)16-s + (−1.01 + 0.849i)17-s + 1.74·18-s + (0.271 − 0.962i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.75811 - 0.888122i\)
\(L(\frac12)\) \(\approx\) \(3.75811 - 0.888122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-1.18 + 4.19i)T \)
good3 \( 1 + (-3.03 - 1.10i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (-1.36 + 2.35i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.18 + 2.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0945 + 0.0344i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (4.17 - 3.50i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.235 - 1.33i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (1.61 + 1.35i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (5.26 - 9.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.18T + 37T^{2} \)
41 \( 1 + (-3.30 - 1.20i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.22 - 6.96i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (6.13 + 5.14i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-1.94 + 11.0i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (9.13 - 7.66i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.0303 - 0.172i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-4.17 - 3.50i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.240 - 1.36i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-11.3 - 4.13i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (3.51 + 1.27i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-4.50 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.26 - 2.64i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-2.26 + 1.89i)T + (16.8 - 95.5i)T^{2} \)
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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

−9.999696861888304412449839843705, −9.128459342084706210311726898670, −8.456879324357710516758035917957, −7.65847308344578263536271118106, −6.74152689704890929971560817772, −5.16051472635002599242079896798, −4.32780415556304689852195374169, −3.60980201895130463466517039387, −2.73107728988361733658458566172, −1.60637105604830816388943734945, 1.95610188120990473741408970896, 2.55196499216346884142281972958, 3.69975562819799026184041951902, 4.67875691897416564507904381181, 5.86779914425796349203449960015, 6.96359757997434428965444506154, 7.68436283695006038001541529353, 8.220821627067200610077054438668, 9.140674467992033269117910015029, 9.571877093513995905645689300857

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