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Properties

Label 2-950-95.27-c1-0-6
Degree $2$
Conductor $950$
Sign $0.822 - 0.569i$
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  + (−0.258 − 0.965i)2-s + (−0.965 + 0.258i)3-s + (−0.866 + 0.499i)4-s + (0.499 + 0.866i)6-s + (−2.44 − 2.44i)7-s + (0.707 + 0.707i)8-s + (−1.73 + i)9-s + (0.707 − 0.707i)12-s + (0.517 − 1.93i)13-s + (−1.73 + 2.99i)14-s + (0.500 − 0.866i)16-s + (−1.67 + 0.448i)17-s + (1.41 + 1.41i)18-s + (2.59 + 3.5i)19-s + (2.99 + 1.73i)21-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.557 + 0.149i)3-s + (−0.433 + 0.249i)4-s + (0.204 + 0.353i)6-s + (−0.925 − 0.925i)7-s + (0.249 + 0.249i)8-s + (−0.577 + 0.333i)9-s + (0.204 − 0.204i)12-s + (0.143 − 0.535i)13-s + (−0.462 + 0.801i)14-s + (0.125 − 0.216i)16-s + (−0.405 + 0.108i)17-s + (0.333 + 0.333i)18-s + (0.596 + 0.802i)19-s + (0.654 + 0.377i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.822 - 0.569i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.822 - 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.589927 + 0.184202i\)
\(L(\frac12)\) \(\approx\) \(0.589927 + 0.184202i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
19 \( 1 + (-2.59 - 3.5i)T \)
good3 \( 1 + (0.965 - 0.258i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (2.44 + 2.44i)T + 7iT^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-0.517 + 1.93i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.67 - 0.448i)T + (14.7 - 8.5i)T^{2} \)
23 \( 1 + (-3.34 - 0.896i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.73 + 3i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 + (2.82 - 2.82i)T - 37iT^{2} \)
41 \( 1 + (-6 - 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.896 - 3.34i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.79 - 6.69i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.59 - 4.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.72 - 2.07i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-12 - 6.92i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.24 + 8.36i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.22 - 1.22i)T - 83iT^{2} \)
89 \( 1 + (4.33 + 7.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.81 - 6.76i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.27763678121068252962499499872, −9.564506355944019788186475854241, −8.570583271005807155514231686204, −7.66958990958976594790059959378, −6.68393336512607238919610765971, −5.72961803236925629645046458493, −4.76591607317236726119224380266, −3.64441949432510936807831562513, −2.82488158900620323320571645911, −1.06679443780472622036014041243, 0.39854930195171868114534590144, 2.44275498090225032607503511862, 3.67314540833866296092125350837, 5.06789047992110892256145656668, 5.76728205347706167640501922657, 6.52583122593242303346153092898, 7.14793788937131067844756278013, 8.397511949246450772401101599737, 9.229685783279670738826125985944, 9.515157430897892298414157678511

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