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Properties

Label 2-95-95.49-c1-0-4
Degree $2$
Conductor $95$
Sign $0.971 + 0.236i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
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  + (−0.747 − 0.431i)2-s + (2.66 + 1.53i)3-s + (−0.627 − 1.08i)4-s + (−0.0476 − 2.23i)5-s + (−1.32 − 2.30i)6-s + 0.566i·7-s + 2.80i·8-s + (3.24 + 5.61i)9-s + (−0.928 + 1.69i)10-s − 1.91·11-s − 3.86i·12-s + (−0.168 + 0.0972i)13-s + (0.244 − 0.423i)14-s + (3.31 − 6.03i)15-s + (−0.0438 + 0.0760i)16-s + (−4.58 − 2.64i)17-s + ⋯
L(s)  = 1  + (−0.528 − 0.305i)2-s + (1.53 + 0.888i)3-s + (−0.313 − 0.543i)4-s + (−0.0213 − 0.999i)5-s + (−0.542 − 0.939i)6-s + 0.214i·7-s + 0.993i·8-s + (1.08 + 1.87i)9-s + (−0.293 + 0.534i)10-s − 0.576·11-s − 1.11i·12-s + (−0.0467 + 0.0269i)13-s + (0.0653 − 0.113i)14-s + (0.855 − 1.55i)15-s + (−0.0109 + 0.0190i)16-s + (−1.11 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.971 + 0.236i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.971 + 0.236i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05986 - 0.127061i\)
\(L(\frac12)\) \(\approx\) \(1.05986 - 0.127061i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.0476 + 2.23i)T \)
19 \( 1 + (-2.36 - 3.65i)T \)
good2 \( 1 + (0.747 + 0.431i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-2.66 - 1.53i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - 0.566iT - 7T^{2} \)
11 \( 1 + 1.91T + 11T^{2} \)
13 \( 1 + (0.168 - 0.0972i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.58 + 2.64i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.92 - 1.68i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.36 + 7.56i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 - 0.955iT - 37T^{2} \)
41 \( 1 + (5.02 - 8.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.27 + 2.46i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.65 + 4.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.10 + 4.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.85 + 3.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.75 - 3.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.50 + 2.02i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.59 - 4.49i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.45 - 4.30i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.31 + 5.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.51iT - 83T^{2} \)
89 \( 1 + (-1.68 - 2.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.1 + 7.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.82376364157888123992325728851, −13.40306262567094854217156024345, −11.64385721849233534386121831270, −10.11650003515366570472016234687, −9.585935894036934225207801001055, −8.647284441799818507186265600317, −7.979369748250766627304543712893, −5.33250589313384565736962288308, −4.18374859928146861510605120322, −2.23931499012859337061225783795, 2.55292522299830712596980426390, 3.79134412088231478567209769547, 6.76396166640658034394064477497, 7.41353006250929391637016558204, 8.375668952359316624803779391580, 9.245370858759835591099369723555, 10.53988319541712012348110338928, 12.25559605215387164397432462995, 13.34611874410358440912540122023, 13.83029675968774332589515807362

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