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Properties

Label 2-115-115.22-c1-0-8
Degree $2$
Conductor $115$
Sign $0.930 + 0.366i$
Analytic cond. $0.918279$
Root an. cond. $0.958269$
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  + (0.562 + 0.562i)2-s + (0.534 − 0.534i)3-s − 1.36i·4-s + (−0.233 − 2.22i)5-s + 0.601·6-s + (−0.567 + 0.567i)7-s + (1.89 − 1.89i)8-s + 2.42i·9-s + (1.11 − 1.38i)10-s + 4.83i·11-s + (−0.730 − 0.730i)12-s + (−3.26 + 3.26i)13-s − 0.638·14-s + (−1.31 − 1.06i)15-s − 0.601·16-s + (1.54 − 1.54i)17-s + ⋯
L(s)  = 1  + (0.397 + 0.397i)2-s + (0.308 − 0.308i)3-s − 0.683i·4-s + (−0.104 − 0.994i)5-s + 0.245·6-s + (−0.214 + 0.214i)7-s + (0.669 − 0.669i)8-s + 0.809i·9-s + (0.354 − 0.437i)10-s + 1.45i·11-s + (−0.210 − 0.210i)12-s + (−0.904 + 0.904i)13-s − 0.170·14-s + (−0.339 − 0.274i)15-s − 0.150·16-s + (0.375 − 0.375i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.930 + 0.366i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1/2),\ 0.930 + 0.366i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30454 - 0.247623i\)
\(L(\frac12)\) \(\approx\) \(1.30454 - 0.247623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.233 + 2.22i)T \)
23 \( 1 + (2.45 + 4.11i)T \)
good2 \( 1 + (-0.562 - 0.562i)T + 2iT^{2} \)
3 \( 1 + (-0.534 + 0.534i)T - 3iT^{2} \)
7 \( 1 + (0.567 - 0.567i)T - 7iT^{2} \)
11 \( 1 - 4.83iT - 11T^{2} \)
13 \( 1 + (3.26 - 3.26i)T - 13iT^{2} \)
17 \( 1 + (-1.54 + 1.54i)T - 17iT^{2} \)
19 \( 1 - 7.21T + 19T^{2} \)
29 \( 1 + 5.91iT - 29T^{2} \)
31 \( 1 + 1.58T + 31T^{2} \)
37 \( 1 + (6.03 - 6.03i)T - 37iT^{2} \)
41 \( 1 - 1.73T + 41T^{2} \)
43 \( 1 + (4.49 + 4.49i)T + 43iT^{2} \)
47 \( 1 + (-0.427 - 0.427i)T + 47iT^{2} \)
53 \( 1 + (6.85 + 6.85i)T + 53iT^{2} \)
59 \( 1 + 7.76iT - 59T^{2} \)
61 \( 1 - 7.62iT - 61T^{2} \)
67 \( 1 + (-1.20 + 1.20i)T - 67iT^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + (-0.559 + 0.559i)T - 73iT^{2} \)
79 \( 1 - 5.66T + 79T^{2} \)
83 \( 1 + (2.15 + 2.15i)T + 83iT^{2} \)
89 \( 1 + 9.48T + 89T^{2} \)
97 \( 1 + (-5.03 + 5.03i)T - 97iT^{2} \)
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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

−13.69564508028140547519502727514, −12.61548943875293458307722146156, −11.77604066201361668890081541924, −10.00515237095619691564553873530, −9.429219472715463309231356735697, −7.84347119330584374057523577814, −6.91135527999743459857227174057, −5.25875597380419593790207028299, −4.56738454801756714733206321779, −1.91345111599671797539715026534, 3.14992461662657701232646384212, 3.51606841080895166037933950084, 5.56555596448381546002921792285, 7.14943867803600015270418582350, 8.123841751783293507144010919974, 9.533112716036470992484521750431, 10.66503933141376466596335469521, 11.62198528178484861294560137977, 12.51367372226587829835193599724, 13.78452118014866251921718309558

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