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Properties

Label 2-195-65.47-c1-0-3
Degree $2$
Conductor $195$
Sign $-0.102 - 0.994i$
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  + 1.58i·2-s + (0.707 − 0.707i)3-s − 0.498·4-s + (−1.18 + 1.89i)5-s + (1.11 + 1.11i)6-s + 0.974·7-s + 2.37i·8-s − 1.00i·9-s + (−2.99 − 1.87i)10-s + (−0.601 + 0.601i)11-s + (−0.352 + 0.352i)12-s + (3.34 + 1.34i)13-s + 1.54i·14-s + (0.502 + 2.17i)15-s − 4.74·16-s + (1.16 − 1.16i)17-s + ⋯
L(s)  = 1  + 1.11i·2-s + (0.408 − 0.408i)3-s − 0.249·4-s + (−0.530 + 0.847i)5-s + (0.456 + 0.456i)6-s + 0.368·7-s + 0.839i·8-s − 0.333i·9-s + (−0.947 − 0.592i)10-s + (−0.181 + 0.181i)11-s + (−0.101 + 0.101i)12-s + (0.927 + 0.373i)13-s + 0.411i·14-s + (0.129 + 0.562i)15-s − 1.18·16-s + (0.282 − 0.282i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.102 - 0.994i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.102 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.915942 + 1.01558i\)
\(L(\frac12)\) \(\approx\) \(0.915942 + 1.01558i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (1.18 - 1.89i)T \)
13 \( 1 + (-3.34 - 1.34i)T \)
good2 \( 1 - 1.58iT - 2T^{2} \)
7 \( 1 - 0.974T + 7T^{2} \)
11 \( 1 + (0.601 - 0.601i)T - 11iT^{2} \)
17 \( 1 + (-1.16 + 1.16i)T - 17iT^{2} \)
19 \( 1 + (-0.691 + 0.691i)T - 19iT^{2} \)
23 \( 1 + (1.66 + 1.66i)T + 23iT^{2} \)
29 \( 1 + 6.37iT - 29T^{2} \)
31 \( 1 + (7.57 + 7.57i)T + 31iT^{2} \)
37 \( 1 - 7.22T + 37T^{2} \)
41 \( 1 + (-3.65 - 3.65i)T + 41iT^{2} \)
43 \( 1 + (-2.71 - 2.71i)T + 43iT^{2} \)
47 \( 1 - 3.25T + 47T^{2} \)
53 \( 1 + (-5.65 + 5.65i)T - 53iT^{2} \)
59 \( 1 + (5.91 + 5.91i)T + 59iT^{2} \)
61 \( 1 - 2.69T + 61T^{2} \)
67 \( 1 + 6.40iT - 67T^{2} \)
71 \( 1 + (-7.17 - 7.17i)T + 71iT^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 - 7.43iT - 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 + (10.6 + 10.6i)T + 89iT^{2} \)
97 \( 1 - 14.4iT - 97T^{2} \)
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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

−13.03735498412087671257531542058, −11.57269629222331617765088162183, −11.10140746981766235820265162826, −9.556300945350815735684291445846, −8.199725216166900990955637354666, −7.68854856360945466631744561996, −6.71204269449770043139731901741, −5.81626281933263534474272241014, −4.11042580030152505151970825078, −2.43737511344949467732538455876, 1.43553267264269678778624167399, 3.24455066430989424921162052613, 4.18106848078363606491202292338, 5.57259429978683466572914578541, 7.40726284113735424826031043304, 8.530461468891344411279042778457, 9.332207092377656513260282223796, 10.57394206408132385706243764881, 11.14567676935699379925531697825, 12.24046542546171122829000392906

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