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Properties

Label 2-950-95.32-c1-0-10
Degree $2$
Conductor $950$
Sign $0.747 - 0.664i$
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  + (0.0871 − 0.996i)2-s + (1.01 + 2.18i)3-s + (−0.984 − 0.173i)4-s + (2.26 − 0.824i)6-s + (1.01 − 0.271i)7-s + (−0.258 + 0.965i)8-s + (−1.80 + 2.15i)9-s + (0.152 + 0.264i)11-s + (−0.624 − 2.32i)12-s + (2.92 + 1.36i)13-s + (−0.181 − 1.03i)14-s + (0.939 + 0.342i)16-s + (1.96 + 0.171i)17-s + (1.99 + 1.99i)18-s + (−1.55 + 4.07i)19-s + ⋯
L(s)  = 1  + (0.0616 − 0.704i)2-s + (0.588 + 1.26i)3-s + (−0.492 − 0.0868i)4-s + (0.925 − 0.336i)6-s + (0.382 − 0.102i)7-s + (−0.0915 + 0.341i)8-s + (−0.603 + 0.718i)9-s + (0.0460 + 0.0797i)11-s + (−0.180 − 0.672i)12-s + (0.811 + 0.378i)13-s + (−0.0486 − 0.275i)14-s + (0.234 + 0.0855i)16-s + (0.475 + 0.0416i)17-s + (0.469 + 0.469i)18-s + (−0.355 + 0.934i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.747 - 0.664i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.747 - 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86686 + 0.709488i\)
\(L(\frac12)\) \(\approx\) \(1.86686 + 0.709488i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0871 + 0.996i)T \)
5 \( 1 \)
19 \( 1 + (1.55 - 4.07i)T \)
good3 \( 1 + (-1.01 - 2.18i)T + (-1.92 + 2.29i)T^{2} \)
7 \( 1 + (-1.01 + 0.271i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.152 - 0.264i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.92 - 1.36i)T + (8.35 + 9.95i)T^{2} \)
17 \( 1 + (-1.96 - 0.171i)T + (16.7 + 2.95i)T^{2} \)
23 \( 1 + (0.601 + 0.858i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (-2.09 - 1.75i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-3.31 - 1.91i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.822 - 0.822i)T - 37iT^{2} \)
41 \( 1 + (2.48 - 6.83i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (5.48 + 3.83i)T + (14.7 + 40.4i)T^{2} \)
47 \( 1 + (-0.0499 - 0.570i)T + (-46.2 + 8.16i)T^{2} \)
53 \( 1 + (-9.03 + 6.32i)T + (18.1 - 49.8i)T^{2} \)
59 \( 1 + (-5.85 + 4.91i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.34 - 13.3i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (5.65 - 0.495i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (1.34 - 0.237i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (-13.4 + 6.26i)T + (46.9 - 55.9i)T^{2} \)
79 \( 1 + (-2.32 - 0.844i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.92 + 7.19i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (7.69 - 2.80i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (1.12 - 12.8i)T + (-95.5 - 16.8i)T^{2} \)
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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

−10.18000072777622456418930470854, −9.512738474200395934442185550998, −8.572082921976763728532634657320, −8.157491632312096704621031709207, −6.65712137915292619981791938041, −5.45032074832956868589682155290, −4.50513248045203535789860823819, −3.81529115818623364363041891241, −2.98038215875731890341234442652, −1.54489493247811139476613535251, 0.970453516264529900699776636405, 2.30747365709264376314508449929, 3.51355856609529777743818417539, 4.81619049385614765374672233539, 5.88807084563200983952834239753, 6.69409374063127287806847855472, 7.40538612353174524263215189647, 8.277815634130093886947741867716, 8.584039690916044974722982958197, 9.698606976013059640293924967687

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