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Properties

Label 2-195-65.49-c1-0-5
Degree $2$
Conductor $195$
Sign $-0.907 - 0.419i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
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  + (1.36 + 2.36i)2-s + (−0.866 + 0.5i)3-s + (−2.72 + 4.71i)4-s + (1.91 + 1.16i)5-s + (−2.36 − 1.36i)6-s + (1.86 − 3.23i)7-s − 9.39·8-s + (0.499 − 0.866i)9-s + (−0.138 + 6.09i)10-s + (−2.02 + 1.17i)11-s − 5.44i·12-s + (1.39 − 3.32i)13-s + 10.1·14-s + (−2.23 − 0.0506i)15-s + (−7.36 − 12.7i)16-s + (2.29 + 1.32i)17-s + ⋯
L(s)  = 1  + (0.964 + 1.67i)2-s + (−0.499 + 0.288i)3-s + (−1.36 + 2.35i)4-s + (0.854 + 0.519i)5-s + (−0.964 − 0.556i)6-s + (0.705 − 1.22i)7-s − 3.32·8-s + (0.166 − 0.288i)9-s + (−0.0436 + 1.92i)10-s + (−0.611 + 0.352i)11-s − 1.57i·12-s + (0.386 − 0.922i)13-s + 2.72·14-s + (−0.577 − 0.0130i)15-s + (−1.84 − 3.19i)16-s + (0.557 + 0.321i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.907 - 0.419i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.907 - 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.364467 + 1.65776i\)
\(L(\frac12)\) \(\approx\) \(0.364467 + 1.65776i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-1.91 - 1.16i)T \)
13 \( 1 + (-1.39 + 3.32i)T \)
good2 \( 1 + (-1.36 - 2.36i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.86 + 3.23i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.02 - 1.17i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.29 - 1.32i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.93 + 1.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.387 + 0.223i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.774 - 1.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.17iT - 31T^{2} \)
37 \( 1 + (-0.797 - 1.38i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.866 - 0.500i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (10.1 + 5.83i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.06T + 47T^{2} \)
53 \( 1 + 8.33iT - 53T^{2} \)
59 \( 1 + (4.61 + 2.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.317 + 0.550i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.06 + 8.76i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.28 - 1.89i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.0493T + 73T^{2} \)
79 \( 1 - 1.93T + 79T^{2} \)
83 \( 1 + 7.63T + 83T^{2} \)
89 \( 1 + (4.56 - 2.63i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.24 - 10.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.33396504128179826987585706951, −12.44741477991645219979850485601, −10.90982336166783477587065006640, −10.03388905979468491632358596956, −8.446825987810223403152081235745, −7.45890790446958181091349814856, −6.62906821453926195623598234797, −5.55280813664689960568667557081, −4.77087699137705510437418256532, −3.45919421855273152571541122171, 1.54776318708462390438840609667, 2.59650826010416239707991695362, 4.49698722770949792243395702509, 5.44676370287862654231128757691, 6.06074373536215336765540807109, 8.549707924984261410934687459441, 9.450685969336748034091013270949, 10.46890247302969981412365869851, 11.44942698537714281764456532820, 12.04126880905656810888289929608

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