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Properties

Label 2-195-195.8-c2-0-10
Degree $2$
Conductor $195$
Sign $-0.990 + 0.135i$
Analytic cond. $5.31336$
Root an. cond. $2.30507$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s + (2.70 + 1.29i)3-s − 4.00·4-s + (−4.94 − 0.707i)5-s + (−3.65 + 7.65i)6-s + 5i·7-s + (5.65 + 7i)9-s + (2.00 − 14.0i)10-s + (−9.19 − 9.19i)11-s + (−10.8 − 5.17i)12-s + 13i·13-s − 14.1·14-s + (−12.4 − 8.31i)15-s − 15.9·16-s + (−13.4 + 13.4i)17-s + (−19.7 + 16i)18-s + ⋯
L(s)  = 1  + 1.41i·2-s + (0.902 + 0.430i)3-s − 1.00·4-s + (−0.989 − 0.141i)5-s + (−0.609 + 1.27i)6-s + 0.714i·7-s + (0.628 + 0.777i)9-s + (0.200 − 1.40i)10-s + (−0.835 − 0.835i)11-s + (−0.902 − 0.430i)12-s + i·13-s − 1.01·14-s + (−0.832 − 0.554i)15-s − 0.999·16-s + (−0.790 + 0.790i)17-s + (−1.09 + 0.888i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.990 + 0.135i$
Analytic conductor: \(5.31336\)
Root analytic conductor: \(2.30507\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1),\ -0.990 + 0.135i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.102147 - 1.50249i\)
\(L(\frac12)\) \(\approx\) \(0.102147 - 1.50249i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.70 - 1.29i)T \)
5 \( 1 + (4.94 + 0.707i)T \)
13 \( 1 - 13iT \)
good2 \( 1 - 2.82iT - 4T^{2} \)
7 \( 1 - 5iT - 49T^{2} \)
11 \( 1 + (9.19 + 9.19i)T + 121iT^{2} \)
17 \( 1 + (13.4 - 13.4i)T - 289iT^{2} \)
19 \( 1 + (-22 + 22i)T - 361iT^{2} \)
23 \( 1 + (-14.8 - 14.8i)T + 529iT^{2} \)
29 \( 1 - 38.1T + 841T^{2} \)
31 \( 1 + (13 + 13i)T + 961iT^{2} \)
37 \( 1 - 15iT - 1.36e3T^{2} \)
41 \( 1 + (24.7 - 24.7i)T - 1.68e3iT^{2} \)
43 \( 1 + (-17 + 17i)T - 1.84e3iT^{2} \)
47 \( 1 - 72.1T + 2.20e3T^{2} \)
53 \( 1 + (-9.19 + 9.19i)T - 2.80e3iT^{2} \)
59 \( 1 + (48.0 - 48.0i)T - 3.48e3iT^{2} \)
61 \( 1 - 67T + 3.72e3T^{2} \)
67 \( 1 + 100T + 4.48e3T^{2} \)
71 \( 1 + (-37.4 + 37.4i)T - 5.04e3iT^{2} \)
73 \( 1 + 76T + 5.32e3T^{2} \)
79 \( 1 - 89iT - 6.24e3T^{2} \)
83 \( 1 - 93.3T + 6.88e3T^{2} \)
89 \( 1 + (-4.94 + 4.94i)T - 7.92e3iT^{2} \)
97 \( 1 + 25T + 9.40e3T^{2} \)
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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

−13.28868445223659030958094554047, −11.77257068392913005243677451843, −10.83713741129670159461412427777, −9.097657188866752454971712619584, −8.673767858929156923404660997975, −7.75899826693347072243660340303, −6.87422290070678579695976874908, −5.38504310207685658370972249340, −4.37443223710204944055053514711, −2.80829110743917917383870220929, 0.809907518020263380382782881715, 2.58687963085365855790192409372, 3.50122148280273341279165607823, 4.63379491403698956166341390953, 7.06853647537111344488803599267, 7.70252407730134744849125046187, 8.905905935234667331588780067270, 10.14366368517135418606603618765, 10.67119711996741483897362772942, 11.97851163763056816561553796000

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